Theorem pm13.193 40104

Description: Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)

Assertion

Ref	Expression
pm13.193	⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝑥 = 𝑦))

Proof of Theorem pm13.193

Step	Hyp	Ref	Expression
1		sbequ12 2177	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
2	1	pm5.32ri 568	1 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) ↔ ([𝑦 / 𝑥]𝜑 ∧ 𝑥 = 𝑦))

Syntax hints:   ↔ wb 198   ∧ wa 387  [wsb 2013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-12 2104

This theorem depends on definitions:  df-bi 199  df-an 388  df-ex 1743  df-sb 2014

This theorem is referenced by: (None)