Theorem 2mos 2674

Description: Double "there exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.) (Proof shortened by Wolf Lammen, 21-May-2025.)

Hypothesis

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

Assertion

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

Distinct variable groups:   𝑧,𝑤,𝜑   𝑥,𝑦,𝜓   𝑥,𝑧,𝑤,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑧, 𝑤)

Proof of Theorem 2mos

Step	Hyp	Ref	Expression
1		2mo 2673	. 2 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
2		2mos.1	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
3	2	2sbievw 2132	. . . . . 6 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ 𝜓)
4	3	anbi2i 635	. . . . 5 ⊢ ((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (𝜑 ∧ 𝜓))
5	4	imbi1i 352	. . . 4 ⊢ (((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ((𝜑 ∧ 𝜓) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
6	5	2albii 1852	. . 3 ⊢ (∀𝑧∀𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
7	6	2albii 1852	. 2 ⊢ (∀𝑥∀𝑦∀𝑧∀𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
8	1, 7	bitri 278	1 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567  ∃wex 1811  [wsb 2098

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564

This theorem is used by: (None)