Theorem sbcco2 2468

Description: A composition law for class substitution. Importantly, x may occur free in the class expression substituted for A. (The proof was 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

Proof of Theorem sbcco2

Step	Hyp	Ref	Expression
1		ax-17 1317	. 2 |- ([A / x]ph -> A.y[A / x]ph)
2		sbcco2.1	. . . 4 |- (x = y -> A = B)
3	2	equcoms 1489	. . 3 |- (y = x -> A = B)
4		dfsbcq 2455	. . . 4 |- (A = B -> ([A / x]ph <-> [B / x]ph))
5	4	bicomd 580	. . 3 |- (A = B -> ([B / x]ph <-> [A / x]ph))
6	3, 5	syl 12	. 2 |- (y = x -> ([B / x]ph <-> [A / x]ph))
7	1, 6	sbie 1565	1 |- ([x / y][B / x]ph <-> [A / x]ph)

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298  [wsbc 1534

This theorem is referenced by:  tfinds2 3947

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-8 1306  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-cleq 1877  df-clel 1880  df-sbc 2454

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