Theorem stdpc7 1917

Description: One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1687.) Translated to traditional notation, it can be read: "x = y → (φ(x, x) → φ(x, y)), provided that y is free for x in φ(x, x)." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)

Assertion

Ref	Expression
stdpc7	⊢ (x = y → ([x / y]φ → φ))

Proof of Theorem stdpc7

Step	Hyp	Ref	Expression
1		sbequ2 1650	. 2 ⊢ (y = x → ([x / y]φ → φ))
2	1	equcoms 1681	1 ⊢ (x = y → ([x / y]φ → φ))

Syntax hints:   → wi 4  [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

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

This theorem is referenced by:  ax16ALT2  2048  sbequi  2059  sb5rf  2090  sb8  2092