Theorem pm14.123b

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

	Ref	Expression
Assertion	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 ) ) )

Proof

Step	Hyp	Ref	Expression
1		2sbc5g	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ∃ 𝑧 ∃ 𝑤 ( ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ∧ 𝜑 ) ↔ [ 𝐴 / 𝑧 ] [ 𝐵 / 𝑤 ] 𝜑 ) )
2	1	adantr	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ ∀ 𝑧 ∀ 𝑤 ( 𝜑 → ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) ) → ( ∃ 𝑧 ∃ 𝑤 ( ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ∧ 𝜑 ) ↔ [ 𝐴 / 𝑧 ] [ 𝐵 / 𝑤 ] 𝜑 ) )
3		nfa1	⊢ Ⅎ 𝑧 ∀ 𝑧 ∀ 𝑤 ( 𝜑 → ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) )
4		nfa2	⊢ Ⅎ 𝑤 ∀ 𝑧 ∀ 𝑤 ( 𝜑 → ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) )
5		simpr	⊢ ( ( ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ∧ 𝜑 ) → 𝜑 )
6		2sp	⊢ ( ∀ 𝑧 ∀ 𝑤 ( 𝜑 → ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) → ( 𝜑 → ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) )
7	6	ancrd	⊢ ( ∀ 𝑧 ∀ 𝑤 ( 𝜑 → ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) → ( 𝜑 → ( ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ∧ 𝜑 ) ) )
8	5 7	impbid2	⊢ ( ∀ 𝑧 ∀ 𝑤 ( 𝜑 → ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) → ( ( ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ∧ 𝜑 ) ↔ 𝜑 ) )
9	4 8	exbid	⊢ ( ∀ 𝑧 ∀ 𝑤 ( 𝜑 → ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) → ( ∃ 𝑤 ( ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ∧ 𝜑 ) ↔ ∃ 𝑤 𝜑 ) )
10	3 9	exbid	⊢ ( ∀ 𝑧 ∀ 𝑤 ( 𝜑 → ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) → ( ∃ 𝑧 ∃ 𝑤 ( ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ∧ 𝜑 ) ↔ ∃ 𝑧 ∃ 𝑤 𝜑 ) )
11	10	adantl	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ ∀ 𝑧 ∀ 𝑤 ( 𝜑 → ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) ) → ( ∃ 𝑧 ∃ 𝑤 ( ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ∧ 𝜑 ) ↔ ∃ 𝑧 ∃ 𝑤 𝜑 ) )
12	2 11	bitr3d	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ∧ ∀ 𝑧 ∀ 𝑤 ( 𝜑 → ( 𝑧 = 𝐴 ∧ 𝑤 = 𝐵 ) ) ) → ( [ 𝐴 / 𝑧 ] [ 𝐵 / 𝑤 ] 𝜑 ↔ ∃ 𝑧 ∃ 𝑤 𝜑 ) )
13	12	pm5.32da	⊢ ( ( 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 ) ) )