Theorem sbccomieg 37357

Description: Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypothesis

Ref	Expression
sbccomieg.1	|- ( a = A -> B = C )

Assertion

Ref	Expression
sbccomieg	|- ( [. A / a ]. [. B / b ]. ph <-> [. C / b ]. [. A / a ]. ph )

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

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

Proof of Theorem sbccomieg

Step	Hyp	Ref	Expression
1		sbcex 3445	. 2 |- ( [. A / a ]. [. B / b ]. ph -> A e. _V )
2		spesbc 3521	. . 3 |- ( [. C / b ]. [. A / a ]. ph -> E. b [. A / a ]. ph )
3		sbcex 3445	. . . 4 |- ( [. A / a ]. ph -> A e. _V )
4	3	exlimiv 1858	. . 3 |- ( E. b [. A / a ]. ph -> A e. _V )
5	2, 4	syl 17	. 2 |- ( [. C / b ]. [. A / a ]. ph -> A e. _V )
6		nfcv 2764	. . . 4 |- F/_ a C
7		nfsbc1v 3455	. . . 4 |- F/ a [. A / a ]. ph
8	6, 7	nfsbc 3457	. . 3 |- F/ a [. C / b ]. [. A / a ]. ph
9		sbccomieg.1	. . . 4 |- ( a = A -> B = C )
10		sbceq1a 3446	. . . 4 |- ( a = A -> ( ph <-> [. A / a ]. ph ) )
11	9, 10	sbceqbid 3442	. . 3 |- ( a = A -> ( [. B / b ]. ph <-> [. C / b ]. [. A / a ]. ph ) )
12	8, 11	sbciegf 3467	. 2 |- ( A e. _V -> ( [. A / a ]. [. B / b ]. ph <-> [. C / b ]. [. A / a ]. ph ) )
13	1, 5, 12	pm5.21nii 368	1 |- ( [. A / a ]. [. B / b ]. ph <-> [. C / b ]. [. A / a ]. ph )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   _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-3an 1039  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:  2rexfrabdioph  37360  3rexfrabdioph  37361  4rexfrabdioph  37362  6rexfrabdioph  37363  7rexfrabdioph  37364