Theorem 2mos 2652

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 2651	. 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 2096	. . . . . 6 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ 𝜓)
4	3	anbi2i 622	. . . . 5 ⊢ ((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (𝜑 ∧ 𝜓))
5	4	imbi1i 349	. . . 4 ⊢ (((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ((𝜑 ∧ 𝜓) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
6	5	2albii 1818	. . 3 ⊢ (∀𝑧∀𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
7	6	2albii 1818	. 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 275	1 ⊢ (∃𝑧∃𝑤∀𝑥∀𝑦(𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  ∃wex 1777  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543

This theorem is referenced by: (None)