Theorem sbcco2 2780

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 2762	. 2 |- ([x / y] [.B / x ].ph <-> [.x / y ]. [.B / x ].ph)
2		nfv 1418	. . 3 |- F/y [.A / x ].ph
3		sbcco2.1	. . . . 5 |- (x = y -> A = B)
4	3	equcoms 1591	. . . 4 |- (y = x -> A = B)
5		dfsbcq 2760	. . . . 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 1671	. 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 1242  [wsb 1642   [.wsbc 2758

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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-sbc 2759

This theorem is referenced by: (None)