Theorem sbcco3g 2903

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 2899	. 2 |- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )
2		elex 2566	. . 3 |- ( A e. V -> A e. _V )
3		nfcvd 2179	. . . 4 |- ( A e. _V -> F/_ x C )
4		sbcco3g.1	. . . 4 |- ( x = A -> B = C )
5	3, 4	csbiegf 2890	. . 3 |- ( A e. _V -> [_ A / x ]_ B = C )
6		dfsbcq 2766	. . 3 |- ( [_ A / x ]_ B = C -> ( [. [_ A / x ]_ B / y ]. ph <-> [. C / y ]. ph ) )
7	2, 5, 6	3syl 17	. 2 |- ( A e. V -> ( [. [_ A / x ]_ B / y ]. ph <-> [. C / y ]. ph ) )
8	1, 7	bitrd 177	1 |- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. C / y ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   _Vcvv 2557   [.wsbc 2764   [_csb 2852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853

This theorem is referenced by:  fzshftral  8970