Theorem sbequ12r 1920

Description: An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Assertion

Ref	Expression
sbequ12r	⊢ (x = y → ([x / y]φ ↔ φ))

Proof of Theorem sbequ12r

Step	Hyp	Ref	Expression
1		sbequ12 1919	. . 3 ⊢ (y = x → (φ ↔ [x / y]φ))
2	1	bicomd 192	. 2 ⊢ (y = x → ([x / y]φ ↔ φ))
3	2	equcoms 1681	1 ⊢ (x = y → ([x / y]φ ↔ 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:  sbidm  2085  abbi  2464  opeliunxp  4821