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Theorem 2rexsb 43176
Description: An equivalent expression for double restricted existence, analogous to rexsb 43174. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
Assertion
Ref Expression
2rexsb (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵𝑥𝑦((𝑥 = 𝑧𝑦 = 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
StepHypRef Expression
1 rexsb 43174 . . . 4 (∃𝑦𝐵 𝜑 ↔ ∃𝑤𝐵𝑦(𝑦 = 𝑤𝜑))
21rexbii 3244 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑤𝐵𝑦(𝑦 = 𝑤𝜑))
3 rexcom 3352 . . 3 (∃𝑥𝐴𝑤𝐵𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑤𝐵𝑥𝐴𝑦(𝑦 = 𝑤𝜑))
42, 3bitri 276 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑤𝐵𝑥𝐴𝑦(𝑦 = 𝑤𝜑))
5 rexsb 43174 . . . . 5 (∃𝑥𝐴𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑧𝐴𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)))
6 impexp 451 . . . . . . . . 9 (((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ (𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
76albii 1811 . . . . . . . 8 (∀𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
8 19.21v 1931 . . . . . . . 8 (∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)) ↔ (𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)))
97, 8bitr2i 277 . . . . . . 7 ((𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)) ↔ ∀𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
109albii 1811 . . . . . 6 (∀𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)) ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1110rexbii 3244 . . . . 5 (∃𝑧𝐴𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)) ↔ ∃𝑧𝐴𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
125, 11bitri 276 . . . 4 (∃𝑥𝐴𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑧𝐴𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1312rexbii 3244 . . 3 (∃𝑤𝐵𝑥𝐴𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑤𝐵𝑧𝐴𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
14 rexcom 3352 . . 3 (∃𝑤𝐵𝑧𝐴𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∃𝑧𝐴𝑤𝐵𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1513, 14bitri 276 . 2 (∃𝑤𝐵𝑥𝐴𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑧𝐴𝑤𝐵𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
164, 15bitri 276 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526  wrex 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-10 2136  ax-11 2151  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ex 1772  df-nf 1776  df-sb 2061  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141
This theorem is referenced by: (None)
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