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Metamath Proof Explorer

Theorem pm13.14

Description: Theorem *13.14 in WhiteheadRussell p. 178. (Contributed by Andrew Salmon, 3-Jun-2011)

Ref Expression
Assertion pm13.14 ( ( [ 𝐴 / 𝑥 ] 𝜑 ∧ ¬ 𝜑 ) → 𝑥 ≠ 𝐴 )

Proof

Step Hyp Ref Expression
1 sbceq1a ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ [ 𝐴 / 𝑥 ] 𝜑 ) )
2 1 biimprcd ⊢ ( [ 𝐴 / 𝑥 ] 𝜑 → ( 𝑥 = 𝐴 → 𝜑 ) )
3 2 necon3bd ⊢ ( [ 𝐴 / 𝑥 ] 𝜑 → ( ¬ 𝜑 → 𝑥 ≠ 𝐴 ) )
4 3 imp ⊢ ( ( [ 𝐴 / 𝑥 ] 𝜑 ∧ ¬ 𝜑 ) → 𝑥 ≠ 𝐴 )