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Metamath Proof Explorer

Theorem pm14.123a

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

Ref Expression
Assertion 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 ) ) )

Proof

Step Hyp Ref Expression
1 2albiim ⊢ ( ∀ 𝑧 ∀ 𝑤 ( 𝜑 ↔ ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) ↔ ( ∀ 𝑧 ∀ 𝑤 ( 𝜑 → ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) ∧ ∀ 𝑧 ∀ 𝑤 ( ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) → 𝜑 ) ) )
2 2sbc6g ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∀ 𝑧 ∀ 𝑤 ( ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) → 𝜑 ) ↔ [ 𝐴 / 𝑧 ] [ 𝐵 / 𝑤 ] 𝜑 ) )
3 2 anbi2d ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( ∀ 𝑧 ∀ 𝑤 ( 𝜑 → ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) ∧ ∀ 𝑧 ∀ 𝑤 ( ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) → 𝜑 ) ) ↔ ( ∀ 𝑧 ∀ 𝑤 ( 𝜑 → ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) ∧ [ 𝐴 / 𝑧 ] [ 𝐵 / 𝑤 ] 𝜑 ) ) )
4 1 3 bitrid ⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∀ 𝑧 ∀ 𝑤 ( 𝜑 ↔ ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) ↔ ( ∀ 𝑧 ∀ 𝑤 ( 𝜑 → ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) ∧ [ 𝐴 / 𝑧 ] [ 𝐵 / 𝑤 ] 𝜑 ) ) )