Theorem 2mos 2540

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

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 2539	. 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		nfv 1830	. . . . . . 7 ⊢ Ⅎ𝑥𝜓
3		2mos.1	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝜑 ↔ 𝜓))
4	3	sbiedv 2398	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ([𝑤 / 𝑦]𝜑 ↔ 𝜓))
5	2, 4	sbie 2396	. . . . . 6 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ 𝜓)
6	5	anbi2i 726	. . . . 5 ⊢ ((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (𝜑 ∧ 𝜓))
7	6	imbi1i 338	. . . 4 ⊢ (((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ((𝜑 ∧ 𝜓) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
8	7	2albii 1738	. . 3 ⊢ (∀𝑧∀𝑤((𝜑 ∧ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∀𝑧∀𝑤((𝜑 ∧ 𝜓) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))
9	8	2albii 1738	. 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)))
10	1, 9	bitri 263	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 195   ∧ wa 383  ∀wal 1473  ∃wex 1695  [wsb 1867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463

This theorem is referenced by: (None)