Theorem sbcex 1465

Description: Move existential quantifier in and out of class substitution.

Assertion

Ref	Expression
sbcex	|- (A e. B -> ([A / y]E.xph <-> E.x[A / y]ph))

Distinct variable group(s):   x,A   x,y

Proof of Theorem sbcex

Step	Hyp	Ref	Expression
1		dfsbcq 1442	. 2 |- (z = A -> ([z / y]E.xph <-> [A / y]E.xph))
2		dfsbcq 1442	. . 3 |- (z = A -> ([z / y]ph <-> [A / y]ph))
3	2	biexdv 936	. 2 |- (z = A -> (E.x[z / y]ph <-> E.x[A / y]ph))
4		sbex 998	. 2 |- ([z / y]E.xph <-> E.x[z / y]ph)
5	1, 3, 4	vtoclbg 1384	1 |- (A e. B -> ([A / y]E.xph <-> E.x[A / y]ph))

Syntax hints:   -> wi 2   <-> wb 127  E.wex 678  [wsb 852   = wceq 1091   e. wcel 1092  [wsbc 1440

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-sbc 1441

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