Theorem 2rexsb 48093

Description: An equivalent expression for double restricted existence, analogous to rexsb 48091. (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 48091	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑤 ∈ 𝐵 ∀𝑦(𝑦 = 𝑤 → 𝜑))
2	1	rexbii 3109	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑦(𝑦 = 𝑤 → 𝜑))
3		rexcom 3291	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑦(𝑦 = 𝑤 → 𝜑) ↔ ∃𝑤 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝑤 → 𝜑))
4	2, 3	bitri 278	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑤 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝑤 → 𝜑))
5		rexsb 48091	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝑤 → 𝜑) ↔ ∃𝑧 ∈ 𝐴 ∀𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜑)))
6		impexp 456	. . . . . . . . 9 ⊢ (((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑) ↔ (𝑥 = 𝑧 → (𝑦 = 𝑤 → 𝜑)))
7	6	albii 1852	. . . . . . . 8 ⊢ (∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑) ↔ ∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤 → 𝜑)))
8		19.21v 1972	. . . . . . . 8 ⊢ (∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤 → 𝜑)) ↔ (𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜑)))
9	7, 8	bitr2i 279	. . . . . . 7 ⊢ ((𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜑)) ↔ ∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
10	9	albii 1852	. . . . . 6 ⊢ (∀𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜑)) ↔ ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
11	10	rexbii 3109	. . . . 5 ⊢ (∃𝑧 ∈ 𝐴 ∀𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤 → 𝜑)) ↔ ∃𝑧 ∈ 𝐴 ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
12	5, 11	bitri 278	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝑤 → 𝜑) ↔ ∃𝑧 ∈ 𝐴 ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
13	12	rexbii 3109	. . 3 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝑤 → 𝜑) ↔ ∃𝑤 ∈ 𝐵 ∃𝑧 ∈ 𝐴 ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
14		rexcom 3291	. . 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 278	. 2 ⊢ (∃𝑤 ∈ 𝐵 ∃𝑥 ∈ 𝐴 ∀𝑦(𝑦 = 𝑤 → 𝜑) ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))
16	4, 15	bitri 278	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥∀𝑦((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-10 2178  ax-11 2194  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087

This theorem is used by: (None)