Theorem 2sb5rf 1338

Description: Reversed double substitution.

Hypotheses

Ref	Expression
2sb5rf.1	|- (ph -> A.zph)
2sb5rf.2	|- (ph -> A.wph)

Assertion

Ref	Expression
2sb5rf	|- (ph <-> E.zE.w((z = x /\ w = y) /\ [z / x][w / y]ph))

Distinct variable groups:   x,y   x,w   y,z   z,w

Proof of Theorem 2sb5rf

Step	Hyp	Ref	Expression
1		2sb5rf.1	. . 3 |- (ph -> A.zph)
2	1	sb5rf 1259	. 2 |- (ph <-> E.z(z = x /\ [z / x]ph))
3		19.42v 1308	. . . 4 |- (E.w(z = x /\ (w = y /\ [w / y][z / x]ph)) <-> (z = x /\ E.w(w = y /\ [w / y][z / x]ph)))
4		sbcom2 1334	. . . . . . 7 |- ([z / x][w / y]ph <-> [w / y][z / x]ph)
5	4	anbi2i 480	. . . . . 6 |- (((z = x /\ w = y) /\ [z / x][w / y]ph) <-> ((z = x /\ w = y) /\ [w / y][z / x]ph))
6		anass 439	. . . . . 6 |- (((z = x /\ w = y) /\ [w / y][z / x]ph) <-> (z = x /\ (w = y /\ [w / y][z / x]ph)))
7	5, 6	bitr 173	. . . . 5 |- (((z = x /\ w = y) /\ [z / x][w / y]ph) <-> (z = x /\ (w = y /\ [w / y][z / x]ph)))
8	7	exbii 1051	. . . 4 |- (E.w((z = x /\ w = y) /\ [z / x][w / y]ph) <-> E.w(z = x /\ (w = y /\ [w / y][z / x]ph)))
9		2sb5rf.2	. . . . . . 7 |- (ph -> A.wph)
10	9	hbsb 1333	. . . . . 6 |- ([z / x]ph -> A.w[z / x]ph)
11	10	sb5rf 1259	. . . . 5 |- ([z / x]ph <-> E.w(w = y /\ [w / y][z / x]ph))
12	11	anbi2i 480	. . . 4 |- ((z = x /\ [z / x]ph) <-> (z = x /\ E.w(w = y /\ [w / y][z / x]ph)))
13	3, 8, 12	3bitr4r 184	. . 3 |- ((z = x /\ [z / x]ph) <-> E.w((z = x /\ w = y) /\ [z / x][w / y]ph))
14	13	exbii 1051	. 2 |- (E.z(z = x /\ [z / x]ph) <-> E.zE.w((z = x /\ w = y) /\ [z / x][w / y]ph))
15	2, 14	bitr 173	1 |- (ph <-> E.zE.w((z = x /\ w = y) /\ [z / x][w / y]ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956  E.wex 980  [wsbc 1170

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-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172

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