Theorem sbequ12a 1921

Description: An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem sbequ12a

Step	Hyp	Ref	Expression
1		sbequ12 1919	. 2 ⊢ (x = y → (φ ↔ [y / x]φ))
2		sbequ12 1919	. . 3 ⊢ (y = x → (φ ↔ [x / y]φ))
3	2	equcoms 1681	. 2 ⊢ (x = y → (φ ↔ [x / y]φ))
4	1, 3	bitr3d 246	1 ⊢ (x = y → ([y / x]φ ↔ [x / y]φ))

Syntax hints:   → wi 4   ↔ wb 176  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649

This theorem is referenced by:  sbco3  2088