Theorem 2exsb 2375

Description: An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.) (Proof shortened by Wolf Lammen, 30-Sep-2018.)

Assertion

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

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

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem 2exsb

Step	Hyp	Ref	Expression
1		nfv 1911	. . 3 ⊢ Ⅎ𝑤𝜑
2		nfv 1911	. . 3 ⊢ Ⅎ𝑧𝜑
3	1, 2	2sb8ev 2371	. 2 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
4		2sb6 2090	. . 3 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
5	4	2exbii 1845	. 2 ⊢ (∃𝑧∃𝑤[𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
6	3, 5	bitri 277	1 ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑧∃𝑤∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531  ∃wex 1776  [wsb 2065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-10 2141  ax-11 2157  ax-12 2173

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1777  df-nf 1781  df-sb 2066

This theorem is referenced by:  2eu6  2740