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Theorem 2rexsb 41676
Description: An equivalent expression for double restricted existence, analogous to rexsb 41674. (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 41674 . . . 4 (∃𝑦𝐵 𝜑 ↔ ∃𝑤𝐵𝑦(𝑦 = 𝑤𝜑))
21rexbii 3179 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑤𝐵𝑦(𝑦 = 𝑤𝜑))
3 rexcom 3237 . . 3 (∃𝑥𝐴𝑤𝐵𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑤𝐵𝑥𝐴𝑦(𝑦 = 𝑤𝜑))
42, 3bitri 264 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑤𝐵𝑥𝐴𝑦(𝑦 = 𝑤𝜑))
5 rexsb 41674 . . . . 5 (∃𝑥𝐴𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑧𝐴𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)))
6 impexp 461 . . . . . . . . 9 (((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ (𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
76albii 1896 . . . . . . . 8 (∀𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
8 19.21v 2017 . . . . . . . 8 (∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)) ↔ (𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)))
97, 8bitr2i 265 . . . . . . 7 ((𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)) ↔ ∀𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
109albii 1896 . . . . . 6 (∀𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)) ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1110rexbii 3179 . . . . 5 (∃𝑧𝐴𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)) ↔ ∃𝑧𝐴𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
125, 11bitri 264 . . . 4 (∃𝑥𝐴𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑧𝐴𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1312rexbii 3179 . . 3 (∃𝑤𝐵𝑥𝐴𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑤𝐵𝑧𝐴𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
14 rexcom 3237 . . 3 (∃𝑤𝐵𝑧𝐴𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∃𝑧𝐴𝑤𝐵𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1513, 14bitri 264 . 2 (∃𝑤𝐵𝑥𝐴𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑧𝐴𝑤𝐵𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
164, 15bitri 264 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630  wrex 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056
This theorem is referenced by: (None)
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