Theorem sbequ12a 867

Description: An equality theorem for substitution.

Assertion

Ref	Expression
sbequ12a	⊢ (x = y → ([y / x]φ ↔ [x / y]φ))

Proof of Theorem sbequ12a

Step	Hyp	Ref	Expression
1		sbequ12 865	. 2 ⊢ (x = y → (φ ↔ [y / x]φ))
2		sbequ12 865	. . 3 ⊢ (y = x → (φ ↔ [x / y]φ))
3	2	eqcoms 813	. 2 ⊢ (x = y → (φ ↔ [x / y]φ))
4	1, 3	bitr3d 408	1 ⊢ (x = y → ([y / x]φ ↔ [x / y]φ))

Syntax hints:   → wi 2   ↔ wb 127   = weq 797  [wsb 852

This theorem is referenced by:  sbco3 915

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-8 798  ax-9 799  ax-12 802

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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