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Theorem 2rexrsb 46511
Description: An equivalent expression for double restricted existence, analogous to 2exsb 2351. (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 46509 . . . 4 (∃𝑦𝐵 𝜑 ↔ ∃𝑤𝐵𝑦𝐵 (𝑦 = 𝑤𝜑))
21rexbii 3091 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑤𝐵𝑦𝐵 (𝑦 = 𝑤𝜑))
3 rexcom 3285 . . 3 (∃𝑥𝐴𝑤𝐵𝑦𝐵 (𝑦 = 𝑤𝜑) ↔ ∃𝑤𝐵𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑))
42, 3bitri 274 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑤𝐵𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑))
5 rexrsb 46509 . . . . 5 (∃𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑) ↔ ∃𝑧𝐴𝑥𝐴 (𝑥 = 𝑧 → ∀𝑦𝐵 (𝑦 = 𝑤𝜑)))
6 impexp 449 . . . . . . . . 9 (((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ (𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
76ralbii 3090 . . . . . . . 8 (∀𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∀𝑦𝐵 (𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
8 r19.21v 3177 . . . . . . . 8 (∀𝑦𝐵 (𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)) ↔ (𝑥 = 𝑧 → ∀𝑦𝐵 (𝑦 = 𝑤𝜑)))
97, 8bitr2i 275 . . . . . . 7 ((𝑥 = 𝑧 → ∀𝑦𝐵 (𝑦 = 𝑤𝜑)) ↔ ∀𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
109ralbii 3090 . . . . . 6 (∀𝑥𝐴 (𝑥 = 𝑧 → ∀𝑦𝐵 (𝑦 = 𝑤𝜑)) ↔ ∀𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1110rexbii 3091 . . . . 5 (∃𝑧𝐴𝑥𝐴 (𝑥 = 𝑧 → ∀𝑦𝐵 (𝑦 = 𝑤𝜑)) ↔ ∃𝑧𝐴𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
125, 11bitri 274 . . . 4 (∃𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑) ↔ ∃𝑧𝐴𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1312rexbii 3091 . . 3 (∃𝑤𝐵𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑) ↔ ∃𝑤𝐵𝑧𝐴𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
14 rexcom 3285 . . 3 (∃𝑤𝐵𝑧𝐴𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1513, 14bitri 274 . 2 (∃𝑤𝐵𝑥𝐴𝑦𝐵 (𝑦 = 𝑤𝜑) ↔ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
164, 15bitri 274 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵𝑥𝐴𝑦𝐵 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wral 3058  wrex 3067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-10 2129  ax-11 2146  ax-12 2166
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-nf 1778  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068
This theorem is referenced by: (None)
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