Theorem sbcco2 2786

Description: A composition law for class substitution. Importantly, x may occur free in the class expression substituted for A. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypothesis

Ref	Expression
sbcco2.1	|- ( x = y -> A = B )

Assertion

Ref	Expression
sbcco2	|- ( [. x / y ]. [. B / x ]. ph <-> [. A / x ]. ph )

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

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

Proof of Theorem sbcco2

Step	Hyp	Ref	Expression
1		sbsbc 2768	. 2 |- ( [ x / y ] [. B / x ]. ph <-> [. x / y ]. [. B / x ]. ph )
2		nfv 1421	. . 3 |- F/ y [. A / x ]. ph
3		sbcco2.1	. . . . 5 |- ( x = y -> A = B )
4	3	equcoms 1594	. . . 4 |- ( y = x -> A = B )
5		dfsbcq 2766	. . . . 5 |- ( A = B -> ( [. A / x ]. ph <-> [. B / x ]. ph ) )
6	5	bicomd 129	. . . 4 |- ( A = B -> ( [. B / x ]. ph <-> [. A / x ]. ph ) )
7	4, 6	syl 14	. . 3 |- ( y = x -> ( [. B / x ]. ph <-> [. A / x ]. ph ) )
8	2, 7	sbie 1674	. 2 |- ( [ x / y ] [. B / x ]. ph <-> [. A / x ]. ph )
9	1, 8	bitr3i 175	1 |- ( [. x / y ]. [. B / x ]. ph <-> [. A / x ]. ph )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   [wsb 1645   [.wsbc 2764

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-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-sbc 2765

This theorem is referenced by: (None)