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 )