Theorem sbcco3g 3106

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 3102	. 2 |- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. [_ A / x ]_ B / y ]. ph ) )
2		elex 2741	. . 3 |- ( A e. V -> A e. _V )
3		nfcvd 2313	. . . 4 |- ( A e. _V -> F/_ x C )
4		sbcco3g.1	. . . 4 |- ( x = A -> B = C )
5	3, 4	csbiegf 3092	. . 3 |- ( A e. _V -> [_ A / x ]_ B = C )
6		dfsbcq 2957	. . 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 187	1 |- ( A e. V -> ( [. A / x ]. [. B / y ]. ph <-> [. C / y ]. ph ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   _Vcvv 2730   [.wsbc 2955   [_csb 3049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050

This theorem is referenced by:  fzshftral  10064