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Theorem 2rexrsb 47052
Description: An equivalent expression for double restricted existence, analogous to 2exsb 2361. (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 47050 . . . 4 (∃𝑦𝐵 𝜑 ↔ ∃𝑤𝐵𝑦𝐵 (𝑦 = 𝑤𝜑))
21rexbii 3092 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑤𝐵𝑦𝐵 (𝑦 = 𝑤𝜑))
3 rexcom 3288 . . 3 (∃𝑥𝐴𝑤𝐵𝑦𝐵 (𝑦 = 𝑤𝜑) ↔ ∃𝑤𝐵𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑))
42, 3bitri 275 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑤𝐵𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑))
5 rexrsb 47050 . . . . 5 (∃𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑) ↔ ∃𝑧𝐴𝑥𝐴 (𝑥 = 𝑧 → ∀𝑦𝐵 (𝑦 = 𝑤𝜑)))
6 impexp 450 . . . . . . . . 9 (((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ (𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
76ralbii 3091 . . . . . . . 8 (∀𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∀𝑦𝐵 (𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
8 r19.21v 3178 . . . . . . . 8 (∀𝑦𝐵 (𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)) ↔ (𝑥 = 𝑧 → ∀𝑦𝐵 (𝑦 = 𝑤𝜑)))
97, 8bitr2i 276 . . . . . . 7 ((𝑥 = 𝑧 → ∀𝑦𝐵 (𝑦 = 𝑤𝜑)) ↔ ∀𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
109ralbii 3091 . . . . . 6 (∀𝑥𝐴 (𝑥 = 𝑧 → ∀𝑦𝐵 (𝑦 = 𝑤𝜑)) ↔ ∀𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1110rexbii 3092 . . . . 5 (∃𝑧𝐴𝑥𝐴 (𝑥 = 𝑧 → ∀𝑦𝐵 (𝑦 = 𝑤𝜑)) ↔ ∃𝑧𝐴𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
125, 11bitri 275 . . . 4 (∃𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑) ↔ ∃𝑧𝐴𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1312rexbii 3092 . . 3 (∃𝑤𝐵𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑) ↔ ∃𝑤𝐵𝑧𝐴𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
14 rexcom 3288 . . 3 (∃𝑤𝐵𝑧𝐴𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1513, 14bitri 275 . 2 (∃𝑤𝐵𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑) ↔ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
164, 15bitri 275 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wral 3059  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-10 2139  ax-11 2155  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069
This theorem is referenced by: (None)
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