Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  2rexsb Structured version   Visualization version   GIF version

Theorem 2rexsb 40471
Description: An equivalent expression for double restricted existence, analogous to rexsb 40469. (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 40469 . . . 4 (∃𝑦𝐵 𝜑 ↔ ∃𝑤𝐵𝑦(𝑦 = 𝑤𝜑))
21rexbii 3034 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝐴𝑤𝐵𝑦(𝑦 = 𝑤𝜑))
3 rexcom 3091 . . 3 (∃𝑥𝐴𝑤𝐵𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑤𝐵𝑥𝐴𝑦(𝑦 = 𝑤𝜑))
42, 3bitri 264 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑤𝐵𝑥𝐴𝑦(𝑦 = 𝑤𝜑))
5 rexsb 40469 . . . . 5 (∃𝑥𝐴𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑧𝐴𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)))
6 impexp 462 . . . . . . . . 9 (((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ (𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
76albii 1744 . . . . . . . 8 (∀𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑) ↔ ∀𝑦(𝑥 = 𝑧 → (𝑦 = 𝑤𝜑)))
8 19.21v 1865 . . . . . . . 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 1744 . . . . . 6 (∀𝑥(𝑥 = 𝑧 → ∀𝑦(𝑦 = 𝑤𝜑)) ↔ ∀𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
1110rexbii 3034 . . . . 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 3034 . . 3 (∃𝑤𝐵𝑥𝐴𝑦(𝑦 = 𝑤𝜑) ↔ ∃𝑤𝐵𝑧𝐴𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) → 𝜑))
14 rexcom 3091 . . 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 384  wal 1478  wrex 2908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator