Theorem sbcco3g 3192

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 3186	. 2 |- (A e. V -> ( [.A / x ]. [.B / y ].ph <-> [.[_A / x]_B / y ].ph))
2		elex 2868	. . 3 |- (A e. V -> A e. _V)
3		nfcvd 2491	. . . 4 |- (A e. _V -> F/_xC)
4		sbcco3g.1	. . . 4 |- (x = A -> B = C)
5	3, 4	csbiegf 3177	. . 3 |- (A e. _V -> [_A / x]_B = C)
6		dfsbcq 3049	. . 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 244	1 |- (A e. V -> ( [.A / x ]. [.B / y ].ph <-> [.C / y ].ph))

Syntax hints:   -> wi 4   <-> wb 176   = wceq 1642   e. wcel 1710  _Vcvv 2860   [.wsbc 3047  [_csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by:  sbcco3gOLD  3193