Theorem sbcco2 1956

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
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-17 973	. 2 |- ([A / x]ph -> A.y[A / x]ph)
2		sbcco2.1	. . . 4 |- (x = y -> A = B)
3		eqcom 1480	. . . 4 |- (y = x <-> x = y)
4		eqcom 1480	. . . 4 |- (B = A <-> A = B)
5	2, 3, 4	3imtr4 219	. . 3 |- (y = x -> B = A)
6		dfsbcq 1946	. . 3 |- (B = A -> ([B / x]ph <-> [A / x]ph))
7	5, 6	syl 10	. 2 |- (y = x -> ([B / x]ph <-> [A / x]ph))
8	1, 7	sbie 1198	1 |- ([x / y][B / x]ph <-> [A / x]ph)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958  [wsbc 1172

This theorem is referenced by:  tfinds2 3171

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-cleq 1472  df-clel 1475  df-sbc 1945

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