Theorem sbcco2 3459

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 3439	. 2 |- ( [ x / y ] [. B / x ]. ph <-> [. x / y ]. [. B / x ]. ph )
2		nfv 1843	. . 3 |- F/ y [. A / x ]. ph
3		sbcco2.1	. . . . 5 |- ( x = y -> A = B )
4	3	equcoms 1947	. . . 4 |- ( y = x -> A = B )
5		dfsbcq 3437	. . . . 5 |- ( A = B -> ( [. A / x ]. ph <-> [. B / x ]. ph ) )
6	5	bicomd 213	. . . 4 |- ( A = B -> ( [. B / x ]. ph <-> [. A / x ]. ph ) )
7	4, 6	syl 17	. . 3 |- ( y = x -> ( [. B / x ]. ph <-> [. A / x ]. ph ) )
8	2, 7	sbie 2408	. 2 |- ( [ x / y ] [. B / x ]. ph <-> [. A / x ]. ph )
9	1, 8	bitr3i 266	1 |- ( [. x / y ]. [. B / x ]. ph <-> [. A / x ]. ph )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   [wsb 1880   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-sbc 3436

This theorem is referenced by:  tfinds2  7063