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	syl5bb	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∀ 𝑧 ∀ 𝑤 ( 𝜑 ↔ ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) ↔ ( ∀ 𝑧 ∀ 𝑤 ( 𝜑 → ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) ∧ [ 𝐴 / 𝑧 ] [ 𝐵 / 𝑤 ] 𝜑 ) ) )