Theorem sbcco2 3012

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 2993	. 2 |- ( [ x / y ] [. B / x ]. ph <-> [. x / y ]. [. B / x ]. ph )
2		nfv 1542	. . 3 |- F/ y [. A / x ]. ph
3		sbcco2.1	. . . . 5 |- ( x = y -> A = B )
4	3	equcoms 1722	. . . 4 |- ( y = x -> A = B )
5		dfsbcq 2991	. . . . 5 |- ( A = B -> ( [. A / x ]. ph <-> [. B / x ]. ph ) )
6	5	bicomd 141	. . . 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 1805	. 2 |- ( [ x / y ] [. B / x ]. ph <-> [. A / x ]. ph )
9	1, 8	bitr3i 186	1 |- ( [. x / y ]. [. B / x ]. ph <-> [. A / x ]. ph )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   [wsb 1776   [.wsbc 2989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-sbc 2990

This theorem is referenced by: (None)