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Metamath Proof Explorer

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 bitrid ⊢ ( 𝐴 ∈ 𝑉 → ( ∀ 𝑥 ( 𝜑 ↔ 𝑥 = 𝐴 ) ↔ ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝐴 ) ∧ [ 𝐴 / 𝑥 ] 𝜑 ) ) )