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Metamath Proof Explorer

Theorem pm13.13b

Description: Theorem *13.13 in WhiteheadRussell p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011)

Ref Expression
Assertion pm13.13b ( ( [ 𝐴 / 𝑥 ] 𝜑 ∧ 𝑥 = 𝐴 ) → 𝜑 )

Proof

Step Hyp Ref Expression
1 sbceq1a ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) )
2 1 biimparc ⊢ ( ( [ 𝐴 / 𝑥 ] 𝜑 ∧ 𝑥 = 𝐴 ) → 𝜑 )