Theorem sbcco2 2973

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 2955	. 2 |- ( [ x / y ] [. B / x ]. ph <-> [. x / y ]. [. B / x ]. ph )
2		nfv 1516	. . 3 |- F/ y [. A / x ]. ph
3		sbcco2.1	. . . . 5 |- ( x = y -> A = B )
4	3	equcoms 1696	. . . 4 |- ( y = x -> A = B )
5		dfsbcq 2953	. . . . 5 |- ( A = B -> ( [. A / x ]. ph <-> [. B / x ]. ph ) )
6	5	bicomd 140	. . . 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 1779	. 2 |- ( [ x / y ] [. B / x ]. ph <-> [. A / x ]. ph )
9	1, 8	bitr3i 185	1 |- ( [. x / y ]. [. B / x ]. ph <-> [. A / x ]. ph )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1343   [wsb 1750   [.wsbc 2951

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-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-sbc 2952

This theorem is referenced by: (None)