Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  equsexv Structured version   Visualization version   GIF version

Theorem equsexv 2262
 Description: Version of equsex 2436 with a disjoint variable condition, which does not require ax-13 2385. See equsexvw 2004 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2261. (Contributed by BJ, 31-May-2019.)
Hypotheses
Ref Expression
equsalv.nf 𝑥𝜓
equsalv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
equsexv (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem equsexv
StepHypRef Expression
1 equsalv.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
21pm5.32i 575 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
32exbii 1841 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜓))
4 ax6ev 1965 . . 3 𝑥 𝑥 = 𝑦
5 equsalv.nf . . . 4 𝑥𝜓
6519.41 2230 . . 3 (∃𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
74, 6mpbiran 705 . 2 (∃𝑥(𝑥 = 𝑦𝜓) ↔ 𝜓)
83, 7bitri 276 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396  ∃wex 1773  Ⅎwnf 1777 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 1904  ax-6 1963  ax-7 2008  ax-12 2169 This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-nf 1778 This theorem is referenced by:  sb5  2270  sb56OLD  2272  equsexhv  2294  cleljustALT2  2378  sb10f  2568  dprd2d2  19088  poimirlem25  34784
 Copyright terms: Public domain W3C validator