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Metamath Proof Explorer

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