Theorem sbcco2OLD 2469

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

Hypothesis

Ref	Expression
sbcco2.1	|- (x = y -> A = B)

Assertion

Ref	Expression
sbcco2OLD	|- ([x / y][B / x]ph <-> [A / x]ph)

Distinct variable groups:   x,y   ph,y   y,A

Proof of Theorem sbcco2OLD

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		eqcom 1886	. . . 4 |- (y = x <-> x = y)
4		eqcom 1886	. . . 4 |- (B = A <-> A = B)
5	2, 3, 4	3imtr4i 236	. . 3 |- (y = x -> B = A)
6		dfsbcq 2455	. . 3 |- (B = A -> ([B / x]ph <-> [A / x]ph))
7	5, 6	syl 12	. 2 |- (y = x -> ([B / x]ph <-> [A / x]ph))
8	1, 7	sbie 1565	1 |- ([x / y][B / x]ph <-> [A / x]ph)

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

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  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

Copyright terms: Public domain