Theorem axrep5 2698

Description: Axiom of Replacement (similar to Axiom Rep of [BellMachover] p. 463). The antecedent tells us ph is analogous to a "function" from x to y (although it is not really a function since it is a wff and not a class). In the consequent we postulate the existence of a set z that corresponds to the "image" of ph restricted to some other set w. The hypothesis says z must not be free in ph.

Hypothesis

Ref	Expression
axrep5.1	|- (ph -> A.zph)

Assertion

Ref	Expression
axrep5	|- (A.x(x e. w -> E.zA.y(ph -> y = z)) -> E.zA.y(y e. z <-> E.x(x e. w /\ ph)))

Distinct variable group:   x,y,z,w

Proof of Theorem axrep5

Step	Hyp	Ref	Expression
1		19.37v 1303	. . . . 5 |- (E.z(x e. w -> A.y(ph -> y = z)) <-> (x e. w -> E.zA.y(ph -> y = z)))
2		impexp 347	. . . . . . . 8 |- (((x e. w /\ ph) -> y = z) <-> (x e. w -> (ph -> y = z)))
3	2	albii 999	. . . . . . 7 |- (A.y((x e. w /\ ph) -> y = z) <-> A.y(x e. w -> (ph -> y = z)))
4		19.21v 1285	. . . . . . 7 |- (A.y(x e. w -> (ph -> y = z)) <-> (x e. w -> A.y(ph -> y = z)))
5	3, 4	bitr2 174	. . . . . 6 |- ((x e. w -> A.y(ph -> y = z)) <-> A.y((x e. w /\ ph) -> y = z))
6	5	exbii 1051	. . . . 5 |- (E.z(x e. w -> A.y(ph -> y = z)) <-> E.zA.y((x e. w /\ ph) -> y = z))
7	1, 6	bitr3 175	. . . 4 |- ((x e. w -> E.zA.y(ph -> y = z)) <-> E.zA.y((x e. w /\ ph) -> y = z))
8	7	albii 999	. . 3 |- (A.x(x e. w -> E.zA.y(ph -> y = z)) <-> A.xE.zA.y((x e. w /\ ph) -> y = z))
9		ax-17 971	. . . . 5 |- (x e. w -> A.z x e. w)
10		axrep5.1	. . . . 5 |- (ph -> A.zph)
11	9, 10	hban 1009	. . . 4 |- ((x e. w /\ ph) -> A.z(x e. w /\ ph))
12	11	axrep4 2697	. . 3 |- (A.xE.zA.y((x e. w /\ ph) -> y = z) -> E.zA.y(y e. z <-> E.x(x e. w /\ (x e. w /\ ph))))
13	8, 12	sylbi 199	. 2 |- (A.x(x e. w -> E.zA.y(ph -> y = z)) -> E.zA.y(y e. z <-> E.x(x e. w /\ (x e. w /\ ph))))
14		anabs5 493	. . . . . 6 |- ((x e. w /\ (x e. w /\ ph)) <-> (x e. w /\ ph))
15	14	exbii 1051	. . . . 5 |- (E.x(x e. w /\ (x e. w /\ ph)) <-> E.x(x e. w /\ ph))
16	15	bibi2i 608	. . . 4 |- ((y e. z <-> E.x(x e. w /\ (x e. w /\ ph))) <-> (y e. z <-> E.x(x e. w /\ ph)))
17	16	albii 999	. . 3 |- (A.y(y e. z <-> E.x(x e. w /\ (x e. w /\ ph))) <-> A.y(y e. z <-> E.x(x e. w /\ ph)))
18	17	exbii 1051	. 2 |- (E.zA.y(y e. z <-> E.x(x e. w /\ (x e. w /\ ph))) <-> E.zA.y(y e. z <-> E.x(x e. w /\ ph)))
19	13, 18	sylib 198	1 |- (A.x(x e. w -> E.zA.y(ph -> y = z)) -> E.zA.y(y e. z <-> E.x(x e. w /\ ph)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980

This theorem is referenced by:  zfrepclf 2699  axsep 2702

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-12 968  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-rep 2693

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981

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