Theorem pm14.122a

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

	Ref	Expression
Assertion	pm14.122a	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝐴 ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ∧ [ 𝐴 / 𝑥 ] 𝜑 ) ) )

Proof

Step	Hyp	Ref	Expression
1		albiim	⊢ ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝐴 ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) )
2		sbc6g	⊢ ( 𝐴 ∈ 𝑉 → ( [ 𝐴 / 𝑥 ] 𝜑 ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) )
3	2	bicomd	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ [ 𝐴 / 𝑥 ] 𝜑 ) )
4	3	anbi2d	⊢ ( 𝐴 ∈ 𝑉 → ( ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ∧ [ 𝐴 / 𝑥 ] 𝜑 ) ) )
5	1 4	syl5bb	⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝐴 ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ∧ [ 𝐴 / 𝑥 ] 𝜑 ) ) )