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

Theorem equsexALT 2422
Description: Alternate proof of equsex 2421. This proves the result directly, instead of as a corollary of equsal 2420 via equs4 2419. Note in particular that only existential quantifiers appear in the proof and that the only step requiring ax-13 2375 is ax6e 2386. This proof mimics that of equsal 2420 (in particular, note that pm5.32i 574, exbii 1845, 19.41 2233, mpbiran 709 correspond respectively to pm5.74i 271, albii 1816, 19.23 2209, a1bi 362). (Contributed by BJ, 20-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
equsal.1 𝑥𝜓
equsal.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
equsexALT (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Proof of Theorem equsexALT
StepHypRef Expression
1 equsal.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
21pm5.32i 574 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
32exbii 1845 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜓))
4 ax6e 2386 . . 3 𝑥 𝑥 = 𝑦
5 equsal.1 . . . 4 𝑥𝜓
6519.41 2233 . . 3 (∃𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
74, 6mpbiran 709 . 2 (∃𝑥(𝑥 = 𝑦𝜓) ↔ 𝜓)
83, 7bitri 275 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wex 1776  wnf 1780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-12 2175  ax-13 2375
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator