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Theorem bj-eu3f 32804
Description: Version of eu3v 2496 where the dv condition is replaced with a non-freeness hypothesis. This is a "backup" of a theorem that used to be in the main part with label "eu3" and was deprecated in favor of eu3v 2496. (Contributed by NM, 8-Jul-1994.) (Proof shortened by BJ, 31-May-2019.)
Hypothesis
Ref Expression
bj-eu3f.1 𝑦𝜑
Assertion
Ref Expression
bj-eu3f (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-eu3f
StepHypRef Expression
1 eu5 2494 . 2 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
2 bj-eu3f.1 . . . 4 𝑦𝜑
32mo2 2477 . . 3 (∃*𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
43anbi2i 729 . 2 ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
51, 4bitri 264 1 (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃𝑦𝑥(𝜑𝑥 = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1479  wex 1702  wnf 1706  ∃!weu 2468  ∃*wmo 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1703  df-nf 1708  df-eu 2472  df-mo 2473
This theorem is referenced by: (None)
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