Theorem pm14.122c

Description: Theorem *14.122 in WhiteheadRussell p. 185. (Contributed by Andrew Salmon, 9-Jun-2011)

	Ref	Expression
Assertion	pm14.122c	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝐴 ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ∧ ∃ 𝑥 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm14.122a	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝐴 ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ∧ [ 𝐴 / 𝑥 ] 𝜑 ) ) )
2		pm14.122b	⊢ ( 𝐴 ∈ 𝑉 → ( ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ∧ [ 𝐴 / 𝑥 ] 𝜑 ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ∧ ∃ 𝑥 𝜑 ) ) )
3	1 2	bitrd	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝐴 ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ∧ ∃ 𝑥 𝜑 ) ) )