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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