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 ( ( 𝐴𝑉𝐵𝑊 ) → ( ∀ 𝑧𝑤 ( 𝜑 ↔ ( 𝑧 = 𝐴𝑤 = 𝐵 ) ) ↔ ( ∀ 𝑧𝑤 ( 𝜑 → ( 𝑧 = 𝐴𝑤 = 𝐵 ) ) ∧ ∃ 𝑧𝑤 𝜑 ) ) )

Proof

Step Hyp Ref Expression
1 pm14.123a ( ( 𝐴𝑉𝐵𝑊 ) → ( ∀ 𝑧𝑤 ( 𝜑 ↔ ( 𝑧 = 𝐴𝑤 = 𝐵 ) ) ↔ ( ∀ 𝑧𝑤 ( 𝜑 → ( 𝑧 = 𝐴𝑤 = 𝐵 ) ) ∧ [ 𝐴 / 𝑧 ] [ 𝐵 / 𝑤 ] 𝜑 ) ) )
2 pm14.123b ( ( 𝐴𝑉𝐵𝑊 ) → ( ( ∀ 𝑧𝑤 ( 𝜑 → ( 𝑧 = 𝐴𝑤 = 𝐵 ) ) ∧ [ 𝐴 / 𝑧 ] [ 𝐵 / 𝑤 ] 𝜑 ) ↔ ( ∀ 𝑧𝑤 ( 𝜑 → ( 𝑧 = 𝐴𝑤 = 𝐵 ) ) ∧ ∃ 𝑧𝑤 𝜑 ) ) )
3 1 2 bitrd ( ( 𝐴𝑉𝐵𝑊 ) → ( ∀ 𝑧𝑤 ( 𝜑 ↔ ( 𝑧 = 𝐴𝑤 = 𝐵 ) ) ↔ ( ∀ 𝑧𝑤 ( 𝜑 → ( 𝑧 = 𝐴𝑤 = 𝐵 ) ) ∧ ∃ 𝑧𝑤 𝜑 ) ) )