Theorem 2rexrsb 44594

Description: An equivalent expression for double restricted existence, analogous to 2exsb 2358. (Contributed by Alexander van der Vekens, 1-Jul-2017.)

Assertion

Ref	Expression
2rexrsb	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑦)

Proof of Theorem 2rexrsb

Step	Hyp	Ref	Expression
1		rexrsb 44592	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑤 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑦 = 𝑤 → 𝜑))
2	1	rexbii 3181	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑦 = 𝑤 → 𝜑))
3		rexcom 3234	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑦 = 𝑤 → 𝜑) ↔ ∃𝑤 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑦 = 𝑤 → 𝜑))
4	2, 3	bitri 274	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑤 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑦 = 𝑤 → 𝜑))
5		rexrsb 44592	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑦 = 𝑤 → 𝜑) ↔ ∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑧 → ∀𝑦 ∈ 𝐵 (𝑦 = 𝑤 → 𝜑)))
6		impexp 451	. . . . . . . . 9 ⊢ (((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑) ↔ (𝑥 = 𝑧 → (𝑦 = 𝑤 → 𝜑)))
7	6	ralbii 3092	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑) ↔ ∀𝑦 ∈ 𝐵 (𝑥 = 𝑧 → (𝑦 = 𝑤 → 𝜑)))
8		r19.21v 3113	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐵 (𝑥 = 𝑧 → (𝑦 = 𝑤 → 𝜑)) ↔ (𝑥 = 𝑧 → ∀𝑦 ∈ 𝐵 (𝑦 = 𝑤 → 𝜑)))
9	7, 8	bitr2i 275	. . . . . . 7 ⊢ ((𝑥 = 𝑧 → ∀𝑦 ∈ 𝐵 (𝑦 = 𝑤 → 𝜑)) ↔ ∀𝑦 ∈ 𝐵 ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
10	9	ralbii 3092	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 = 𝑧 → ∀𝑦 ∈ 𝐵 (𝑦 = 𝑤 → 𝜑)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
11	10	rexbii 3181	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 = 𝑧 → ∀𝑦 ∈ 𝐵 (𝑦 = 𝑤 → 𝜑)) ↔ ∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
12	5, 11	bitri 274	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑦 = 𝑤 → 𝜑) ↔ ∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
13	12	rexbii 3181	. . 3 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑦 = 𝑤 → 𝜑) ↔ ∃𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
14		rexcom 3234	. . 3 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑) ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
15	13, 14	bitri 274	. 2 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑦 = 𝑤 → 𝜑) ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
16	4, 15	bitri 274	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wral 3064  ∃wrex 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070

This theorem is referenced by: (None)