Theorem sbcco3g 2041

Description: Composition of two substitutions.

Hypothesis

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

Assertion

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

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

Proof of Theorem sbcco3g

Step	Hyp	Ref	Expression
1		sbcnestg 2038	. 2 |- ((A e. R /\ A.x B e. S) -> ([A / x][B / y]ph <-> [[_A / x]_B / y]ph))
2		ax-17 971	. . . . . 6 |- (z e. C -> A.x z e. C)
3	2	gen2 983	. . . . 5 |- A.xA.z(z e. C -> A.x z e. C)
4		sbcco3g.1	. . . . . 6 |- (x = A -> B = C)
5	4	ax-gen 963	. . . . 5 |- A.x(x = A -> B = C)
6		csbiegft 2029	. . . . 5 |- ((A e. R /\ A.xA.z(z e. C -> A.x z e. C) /\ A.x(x = A -> B = C)) -> [_A / x]_B = C)
7	3, 5, 6	mp3an23 908	. . . 4 |- (A e. R -> [_A / x]_B = C)
8		dfsbcq 1943	. . . 4 |- ([_A / x]_B = C -> ([[_A / x]_B / y]ph <-> [C / y]ph))
9	7, 8	syl 10	. . 3 |- (A e. R -> ([[_A / x]_B / y]ph <-> [C / y]ph))
10	9	adantr 389	. 2 |- ((A e. R /\ A.x B e. S) -> ([[_A / x]_B / y]ph <-> [C / y]ph))
11	1, 10	bitrd 528	1 |- ((A e. R /\ A.x B e. S) -> ([A / x][B / y]ph <-> [C / y]ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  [wsbc 1170  [_csb 2001

This theorem is referenced by:  fzshftralt 6522

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942  df-csb 2002

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