Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-rexcom4b Structured version   Visualization version   GIF version

Theorem bj-rexcom4b 36849
Description: Remove from rexcom4b 3521 dependency on ax-ext 2711 and ax-13 2380 (and on df-or 847, df-cleq 2732, df-nfc 2895, df-v 3490). The hypothesis uses 𝑉 instead of V (see bj-isseti 36844 for the motivation). Use bj-rexcom4bv 36848 instead when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-rexcom4b.1 𝐵𝑉
Assertion
Ref Expression
bj-rexcom4b (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑦   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)   𝐵(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem bj-rexcom4b
StepHypRef Expression
1 rexcom4a 3298 . 2 (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
2 bj-rexcom4b.1 . . . . 5 𝐵𝑉
32bj-isseti 36844 . . . 4 𝑥 𝑥 = 𝐵
43biantru 529 . . 3 (𝜑 ↔ (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
54rexbii 3100 . 2 (∃𝑦𝐴 𝜑 ↔ ∃𝑦𝐴 (𝜑 ∧ ∃𝑥 𝑥 = 𝐵))
61, 5bitr4i 278 1 (∃𝑥𝑦𝐴 (𝜑𝑥 = 𝐵) ↔ ∃𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wex 1777  wcel 2108  wrex 3076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-11 2158
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-clel 2819  df-rex 3077
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator