Theorem sbc2rexgOLD 37352

Description: Exchange a substitution with two existentials. (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.)

Assertion

Ref	Expression
sbc2rexgOLD	|- ( A e. V -> ( [. A / a ]. E. b e. B E. c e. C ph <-> E. b e. B E. c e. C [. A / a ]. ph ) )

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

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

Proof of Theorem sbc2rexgOLD

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( A e. V -> A e. _V )
2		sbcrexgOLD 37349	. . 3 |- ( A e. _V -> ( [. A / a ]. E. b e. B E. c e. C ph <-> E. b e. B [. A / a ]. E. c e. C ph ) )
3		sbcrexgOLD 37349	. . . 4 |- ( A e. _V -> ( [. A / a ]. E. c e. C ph <-> E. c e. C [. A / a ]. ph ) )
4	3	rexbidv 3052	. . 3 |- ( A e. _V -> ( E. b e. B [. A / a ]. E. c e. C ph <-> E. b e. B E. c e. C [. A / a ]. ph ) )
5	2, 4	bitrd 268	. 2 |- ( A e. _V -> ( [. A / a ]. E. b e. B E. c e. C ph <-> E. b e. B E. c e. C [. A / a ]. ph ) )
6	1, 5	syl 17	1 |- ( A e. V -> ( [. A / a ]. E. b e. B E. c e. C ph <-> E. b e. B E. c e. C [. A / a ]. ph ) )

Syntax hints:   -> wi 4   <-> 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:  sbc4rexgOLD  37354