Theorem sbcco2 1449

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]φ ↔ [A / x]φ)

Distinct variable group(s):   x,y   φ,y   y,A

Proof of Theorem sbcco2

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ ([A / x]φ → ∀y[A / x]φ)
2		sbcco2.1	. . . 4 ⊢ (x = y → A = B)
3		cleqcom 1103	. . . 4 ⊢ (y = x ↔ x = y)
4		cleqcom 1103	. . . 4 ⊢ (B = A ↔ A = B)
5	2, 3, 4	3imtr4 192	. . 3 ⊢ (y = x → B = A)
6		dfsbcq 1442	. . 3 ⊢ (B = A → ([B / x]φ ↔ [A / x]φ))
7	5, 6	syl 12	. 2 ⊢ (y = x → ([B / x]φ ↔ [A / x]φ))
8	1, 7	sbie 904	1 ⊢ ([x / y][B / x]φ ↔ [A / x]φ)

Syntax hints:   → wi 2   ↔ wb 127   = weq 797  [wsb 852   = wceq 1091  [wsbc 1440

This theorem is referenced by:  tfinds2 2405

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-cleq 1097  df-clel 1099  df-sbc 1441

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