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Theorem 2rexrsb 43644
 Description: An equivalent expression for double restricted existence, analogous to 2exsb 2371. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
Assertion
Ref Expression
2rexrsb (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 ((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 2rexrsb
StepHypRef Expression
1 rexrsb 43642 . . . 4 (∃𝑦𝐵 𝜑 ↔ ∃𝑤𝐵𝑦𝐵 (𝑦 = 𝑤𝜑))
21rexbii 3213 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑤𝐵𝑦𝐵 (𝑦 = 𝑤𝜑))
3 rexcom 3311 . . 3 (∃𝑥𝐴𝑤𝐵𝑦𝐵 (𝑦 = 𝑤𝜑) ↔ ∃𝑤𝐵𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑))
42, 3bitri 278 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑤𝐵𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑))
5 rexrsb 43642 . . . . 5 (∃𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑) ↔ ∃𝑧𝐴𝑥𝐴 (𝑥 = 𝑧 → ∀𝑦𝐵 (𝑦 = 𝑤𝜑)))
6 impexp 454 . . . . . . . . 9 (((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ (𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
76ralbii 3136 . . . . . . . 8 (∀𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∀𝑦𝐵 (𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
8 r19.21v 3145 . . . . . . . 8 (∀𝑦𝐵 (𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)) ↔ (𝑥 = 𝑧 → ∀𝑦𝐵 (𝑦 = 𝑤𝜑)))
97, 8bitr2i 279 . . . . . . 7 ((𝑥 = 𝑧 → ∀𝑦𝐵 (𝑦 = 𝑤𝜑)) ↔ ∀𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
109ralbii 3136 . . . . . 6 (∀𝑥𝐴 (𝑥 = 𝑧 → ∀𝑦𝐵 (𝑦 = 𝑤𝜑)) ↔ ∀𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1110rexbii 3213 . . . . 5 (∃𝑧𝐴𝑥𝐴 (𝑥 = 𝑧 → ∀𝑦𝐵 (𝑦 = 𝑤𝜑)) ↔ ∃𝑧𝐴𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
125, 11bitri 278 . . . 4 (∃𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑) ↔ ∃𝑧𝐴𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1312rexbii 3213 . . 3 (∃𝑤𝐵𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑) ↔ ∃𝑤𝐵𝑧𝐴𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
14 rexcom 3311 . . 3 (∃𝑤𝐵𝑧𝐴𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1513, 14bitri 278 . 2 (∃𝑤𝐵𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑) ↔ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
164, 15bitri 278 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wral 3109  ∃wrex 3110 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-10 2143  ax-11 2159  ax-12 2176 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115 This theorem is referenced by: (None)
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