Theorem sbcco3g 3999

Description: Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)

Hypothesis

Ref	Expression
sbcco3g.1	|- ( x = A -> B = C )

Assertion

Ref	Expression
sbcco3g	|- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. C / y ]. ph ) )

Distinct variable groups:   x, A   ph, x   x, C

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

Proof of Theorem sbcco3g

Step	Hyp	Ref	Expression
1		sbcnestg 3997	. 2 |- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )
2		elex 3212	. . 3 |- ( A e. V -> A e. _V )
3		nfcvd 2765	. . . 4 |- ( A e. _V -> F/_ x C )
4		sbcco3g.1	. . . 4 |- ( x = A -> B = C )
5	3, 4	csbiegf 3557	. . 3 |- ( A e. _V -> [_ A / x ]_ B = C )
6		dfsbcq 3437	. . 3 |- ( [_ A / x ]_ B = C -> ( [. [_ A / x ]_ B / y ]. ph <-> [. C / y ]. ph ) )
7	2, 5, 6	3syl 18	. 2 |- ( A e. V -> ( [. [_ A / x ]_ B / y ]. ph <-> [. C / y ]. ph ) )
8	1, 7	bitrd 268	1 |- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. C / y ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   [_csb 3533

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-v 3202  df-sbc 3436  df-csb 3534

This theorem is referenced by:  fzshftral  12428  2rexfrabdioph  37360  3rexfrabdioph  37361  4rexfrabdioph  37362  6rexfrabdioph  37363  7rexfrabdioph  37364