Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-rexcom4a Structured version   Visualization version   GIF version

Theorem bj-rexcom4a 32854
Description: Remove from rexcom4a 3224 dependency on ax-ext 2601 and ax-13 2245 (and on df-or 385, df-sb 1880, df-clab 2608, df-cleq 2614, df-clel 2617, df-nfc 2752, df-v 3200). This proof uses only df-rex 2917 on top of first-order logic. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-rexcom4a (∃𝑥𝑦𝐴 (𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑦)

Proof of Theorem bj-rexcom4a
StepHypRef Expression
1 bj-rexcom4 32853 . 2 (∃𝑦𝐴𝑥(𝜑𝜓) ↔ ∃𝑥𝑦𝐴 (𝜑𝜓))
2 19.42v 1917 . . 3 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))
32rexbii 3039 . 2 (∃𝑦𝐴𝑥(𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
41, 3bitr3i 266 1 (∃𝑥𝑦𝐴 (𝜑𝜓) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wex 1703  wrex 2912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-11 2033
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1704  df-rex 2917
This theorem is referenced by:  bj-rexcom4bv  32855  bj-rexcom4b  32856
  Copyright terms: Public domain W3C validator