Theorem sb7f 1336

Description: This version of dfsb7 1335 does not require that ph and z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 968 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1168 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.)

Hypothesis

Ref	Expression
sb7f.1	|- (ph -> A.zph)

Assertion

Ref	Expression
sb7f	|- ([y / x]ph <-> E.z(z = y /\ E.x(x = z /\ ph)))

Distinct variable groups:   x,z   y,z

Proof of Theorem sb7f

Step	Hyp	Ref	Expression
1		dfsb7 1335	. 2 |- ([y / x]ph <-> E.w(w = y /\ E.x(x = w /\ ph)))
2		ax-17 968	. . . 4 |- (w = y -> A.z w = y)
3		ax-17 968	. . . . . 6 |- (x = w -> A.z x = w)
4		sb7f.1	. . . . . 6 |- (ph -> A.zph)
5	3, 4	hban 1006	. . . . 5 |- ((x = w /\ ph) -> A.z(x = w /\ ph))
6	5	hbex 1003	. . . 4 |- (E.x(x = w /\ ph) -> A.zE.x(x = w /\ ph))
7	2, 6	hban 1006	. . 3 |- ((w = y /\ E.x(x = w /\ ph)) -> A.z(w = y /\ E.x(x = w /\ ph)))
8		ax-17 968	. . 3 |- ((z = y /\ E.x(x = z /\ ph)) -> A.w(z = y /\ E.x(x = z /\ ph)))
9		equequ1 1130	. . . 4 |- (w = z -> (w = y <-> z = y))
10		equequ2 1131	. . . . . 6 |- (w = z -> (x = w <-> x = z))
11	10	anbi1d 615	. . . . 5 |- (w = z -> ((x = w /\ ph) <-> (x = z /\ ph)))
12	11	exbidv 1274	. . . 4 |- (w = z -> (E.x(x = w /\ ph) <-> E.x(x = z /\ ph)))
13	9, 12	anbi12d 626	. . 3 |- (w = z -> ((w = y /\ E.x(x = w /\ ph)) <-> (z = y /\ E.x(x = z /\ ph))))
14	7, 8, 13	cbvex 1162	. 2 |- (E.w(w = y /\ E.x(x = w /\ ph)) <-> E.z(z = y /\ E.x(x = z /\ ph)))
15	1, 14	bitr 173	1 |- ([y / x]ph <-> E.z(z = y /\ E.x(x = z /\ ph)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953  E.wex 977  [wsbc 1166

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168

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