Theorem sbcrexgOLD 37349

Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) Obsolete as of 18-Aug-2018. Use sbcrex 3514 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Distinct variable groups:   y, A   x, B   x, y

Allowed substitution hints:   ph( x, y)   A( x)   B( y)   V( x, y)

Proof of Theorem sbcrexgOLD

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 3438	. 2 |- ( z = A -> ( [ z / x ] E. y e. B ph <-> [. A / x ]. E. y e. B ph ) )
2		dfsbcq2 3438	. . 3 |- ( z = A -> ( [ z / x ] ph <-> [. A / x ]. ph ) )
3	2	rexbidv 3052	. 2 |- ( z = A -> ( E. y e. B [ z / x ] ph <-> E. y e. B [. A / x ]. ph ) )
4		nfcv 2764	. . . 4 |- F/_ x B
5		nfs1v 2437	. . . 4 |- F/ x [ z / x ] ph
6	4, 5	nfrex 3007	. . 3 |- F/ x E. y e. B [ z / x ] ph
7		sbequ12 2111	. . . 4 |- ( x = z -> ( ph <-> [ z / x ] ph ) )
8	7	rexbidv 3052	. . 3 |- ( x = z -> ( E. y e. B ph <-> E. y e. B [ z / x ] ph ) )
9	6, 8	sbie 2408	. 2 |- ( [ z / x ] E. y e. B ph <-> E. y e. B [ z / x ] ph )
10	1, 3, 9	vtoclbg 3267	1 |- ( A e. V -> ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   [wsb 1880   e. wcel 1990   E.wrex 2913   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436

This theorem is referenced by:  2sbcrexOLD  37350  sbc2rexgOLD  37352