Theorem 2mos

Description: Double "there exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005)

	Ref	Expression
Hypothesis	2mos.1	⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 ↔ 𝜓 ) )
Assertion	2mos	⊢ ( ∃ 𝑧 ∃ 𝑤 ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )

Proof

Step	Hyp	Ref	Expression
1		2mos.1	⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝜑 ↔ 𝜓 ) )
2		2mo	⊢ ( ∃ 𝑧 ∃ 𝑤 ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )
3		nfv	⊢ Ⅎ 𝑦 𝑥 = 𝑧
4	3	sbrim	⊢ ( [ 𝑤 / 𝑦 ] ( 𝑥 = 𝑧 → 𝜑 ) ↔ ( 𝑥 = 𝑧 → [ 𝑤 / 𝑦 ] 𝜑 ) )
5	1	expcom	⊢ ( 𝑦 = 𝑤 → ( 𝑥 = 𝑧 → ( 𝜑 ↔ 𝜓 ) ) )
6	5	pm5.74d	⊢ ( 𝑦 = 𝑤 → ( ( 𝑥 = 𝑧 → 𝜑 ) ↔ ( 𝑥 = 𝑧 → 𝜓 ) ) )
7	6	sbievw	⊢ ( [ 𝑤 / 𝑦 ] ( 𝑥 = 𝑧 → 𝜑 ) ↔ ( 𝑥 = 𝑧 → 𝜓 ) )
8	4 7	bitr3i	⊢ ( ( 𝑥 = 𝑧 → [ 𝑤 / 𝑦 ] 𝜑 ) ↔ ( 𝑥 = 𝑧 → 𝜓 ) )
9	8	pm5.74ri	⊢ ( 𝑥 = 𝑧 → ( [ 𝑤 / 𝑦 ] 𝜑 ↔ 𝜓 ) )
10	9	sbievw	⊢ ( [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ↔ 𝜓 )
11	10	anbi2i	⊢ ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) ↔ ( 𝜑 ∧ 𝜓 ) )
12	11	imbi1i	⊢ ( ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )
13	12	2albii	⊢ ( ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )
14	13	2albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ [ 𝑧 / 𝑥 ] [ 𝑤 / 𝑦 ] 𝜑 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )
15	2 14	bitri	⊢ ( ∃ 𝑧 ∃ 𝑤 ∀ 𝑥 ∀ 𝑦 ( 𝜑 → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ∀ 𝑤 ( ( 𝜑 ∧ 𝜓 ) → ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) ) )