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Metamath Proof Explorer

Theorem pm14.123c

Description: Theorem *14.123 in WhiteheadRussell p. 185. (Contributed by Andrew Salmon, 9-Jun-2011)

Ref Expression
Assertion pm14.123c ⊢ A ∈ V ∧ B ∈ W → ∀ z ∀ w φ ↔ z = A ∧ w = B ↔ ∀ z ∀ w φ → z = A ∧ w = B ∧ ∃ z ∃ w φ

Proof

Step Hyp Ref Expression
1 pm14.123a ⊢ A ∈ V ∧ B ∈ W → ∀ z ∀ w φ ↔ z = A ∧ w = B ↔ ∀ z ∀ w φ → z = A ∧ w = B ∧ [˙A / z]˙ [˙B / w]˙ φ
2 pm14.123b ⊢ A ∈ V ∧ B ∈ W → ∀ z ∀ w φ → z = A ∧ w = B ∧ [˙A / z]˙ [˙B / w]˙ φ ↔ ∀ z ∀ w φ → z = A ∧ w = B ∧ ∃ z ∃ w φ
3 1 2 bitrd ⊢ A ∈ V ∧ B ∈ W → ∀ z ∀ w φ ↔ z = A ∧ w = B ↔ ∀ z ∀ w φ → z = A ∧ w = B ∧ ∃ z ∃ w φ