Theorem sbequ12a 1182

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 1180	. 2 ⊢ (x = y → (φ ↔ [y / x]φ))
2		sbequ12 1180	. . 3 ⊢ (y = x → (φ ↔ [x / y]φ))
3	2	equcoms 1129	. 2 ⊢ (x = y → (φ ↔ [x / y]φ))
4	1, 3	bitr3d 529	1 ⊢ (x = y → ([y / x]φ ↔ [x / y]φ))

Syntax hints:   → wi 3   ↔ wb 146   = wceq 955  [wsbc 1169

This theorem is referenced by:  sbco3 1256

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-8 963  ax-12 967  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171

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