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Metamath Proof Explorer

Theorem pm13.193

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

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

Proof

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