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Theorem equsexALT 2418
Description: Alternate proof of equsex 2417. This proves the result directly, instead of as a corollary of equsal 2416 via equs4 2415. Note in particular that only existential quantifiers appear in the proof and that the only step requiring ax-13 2371 is ax6e 2382. This proof mimics that of equsal 2416 (in particular, note that pm5.32i 575, exbii 1850, 19.41 2228, mpbiran 707 correspond respectively to pm5.74i 270, albii 1821, 19.23 2204, 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 575 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
32exbii 1850 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜓))
4 ax6e 2382 . . 3 𝑥 𝑥 = 𝑦
5 equsal.1 . . . 4 𝑥𝜓
6519.41 2228 . . 3 (∃𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
74, 6mpbiran 707 . 2 (∃𝑥(𝑥 = 𝑦𝜓) ↔ 𝜓)
83, 7bitri 274 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wex 1781  wnf 1785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171  ax-13 2371
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-nf 1786
This theorem is referenced by: (None)
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