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Theorem cleljustALT 2365
Description: Alternate proof of cleljust 2115. It is kept here and should not be modified because it is referenced on the Metamath Proof Explorer Home Page (mmset.html) as an example of how disjoint variable conditions are inherited by substitutions. (Contributed by NM, 28-Jan-2004.) (Revised by BJ, 29-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cleljustALT (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem cleljustALT
StepHypRef Expression
1 ax-5 1908 . . 3 (𝑥𝑦 → ∀𝑧 𝑥𝑦)
2 elequ1 2113 . . 3 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
31, 2equsexhv 2291 . 2 (∃𝑧(𝑧 = 𝑥𝑧𝑦) ↔ 𝑥𝑦)
43bicomi 224 1 (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wex 1776
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-8 2108  ax-10 2139  ax-12 2175
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781
This theorem is referenced by: (None)
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