Theorem 2sbcrexOLD 37350

Description: Exchange an existential quantifier with two substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) Obsolete as of 24-Aug-2018. Use csbov123 6687 instead. (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
2sbcrex.1	|- A e. _V
2sbcrex.2	|- B e. _V

Assertion

Ref	Expression
2sbcrexOLD	|- ( [. A / a ]. [. B / b ]. E. c e. C ph <-> E. c e. C [. A / a ]. [. B / b ]. ph )

Distinct variable groups:   A, c   B, c   C, b   a, c   b, c   C, a

Allowed substitution hints:   ph( a, b, c)   A( a, b)   B( a, b)   C( c)

Proof of Theorem 2sbcrexOLD

Step	Hyp	Ref	Expression
1		2sbcrex.2	. . . 4 |- B e. _V
2		sbcrexgOLD 37349	. . . 4 |- ( B e. _V -> ( [. B / b ]. E. c e. C ph <-> E. c e. C [. B / b ]. ph ) )
3	1, 2	ax-mp 5	. . 3 |- ( [. B / b ]. E. c e. C ph <-> E. c e. C [. B / b ]. ph )
4	3	sbcbii 3491	. 2 |- ( [. A / a ]. [. B / b ]. E. c e. C ph <-> [. A / a ]. E. c e. C [. B / b ]. ph )
5		2sbcrex.1	. . 3 |- A e. _V
6		sbcrexgOLD 37349	. . 3 |- ( A e. _V -> ( [. A / a ]. E. c e. C [. B / b ]. ph <-> E. c e. C [. A / a ]. [. B / b ]. ph ) )
7	5, 6	ax-mp 5	. 2 |- ( [. A / a ]. E. c e. C [. B / b ]. ph <-> E. c e. C [. A / a ]. [. B / b ]. ph )
8	4, 7	bitri 264	1 |- ( [. A / a ]. [. B / b ]. E. c e. C ph <-> E. c e. C [. A / a ]. [. B / b ]. ph )

Syntax hints:   <-> wb 196   e. wcel 1990   E.wrex 2913   _Vcvv 3200   [.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: (None)