Theorem 2rexsb 47578

Description: An equivalent expression for double restricted existence, analogous to rexsb 47576. (Contributed by Alexander van der Vekens, 1-Jul-2017.)

Assertion

Ref	Expression
2rexsb	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥∀𝑦((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 2rexsb

Step	Hyp	Ref	Expression
1		rexsb 47576	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑤 ∈ 𝐵 ∀𝑦(𝑦 = 𝑤 → 𝜑))
2	1	rexbii 3088	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑦(𝑦 = 𝑤 → 𝜑))
3		rexcom 3270	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑦(𝑦 = 𝑤 → 𝜑) ↔ ∃𝑤 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝑤 → 𝜑))
4	2, 3	bitri 277	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑤 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝑤 → 𝜑))
5		rexsb 47576	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝑤 → 𝜑) ↔ ∃𝑧 ∈ 𝐴 ∀𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜑)))
6		impexp 452	. . . . . . . . 9 ⊢ (((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑) ↔ (𝑥 = 𝑧 → (𝑦 = 𝑤 → 𝜑)))
7	6	albii 1827	. . . . . . . 8 ⊢ (∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑) ↔ ∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤 → 𝜑)))
8		19.21v 1947	. . . . . . . 8 ⊢ (∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤 → 𝜑)) ↔ (𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜑)))
9	7, 8	bitr2i 278	. . . . . . 7 ⊢ ((𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜑)) ↔ ∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
10	9	albii 1827	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜑)) ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
11	10	rexbii 3088	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 ∀𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜑)) ↔ ∃𝑧 ∈ 𝐴 ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
12	5, 11	bitri 277	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝑤 → 𝜑) ↔ ∃𝑧 ∈ 𝐴 ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
13	12	rexbii 3088	. . 3 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝑤 → 𝜑) ↔ ∃𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
14		rexcom 3270	. . 3 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑) ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
15	13, 14	bitri 277	. 2 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝑤 → 𝜑) ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
16	4, 15	bitri 277	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397  ∀wal 1546  ∃wrex 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-10 2154  ax-11 2170  ax-12 2191

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-ex 1788  df-nf 1792  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066

This theorem is referenced by: (None)