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Theorem equsexv 2106
Description: Version of equsex 2292 with a dv condition, which does not require ax-13 2245. (Contributed by BJ, 31-May-2019.)
Hypotheses
Ref Expression
equsexv.nf 𝑥𝜓
equsexv.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 equsexv.1 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
21pm5.32i 668 . . 3 ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑦𝜓))
32exbii 1771 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑥(𝑥 = 𝑦𝜓))
4 ax6ev 1887 . . 3 𝑥 𝑥 = 𝑦
5 equsexv.nf . . . 4 𝑥𝜓
6519.41 2101 . . 3 (∃𝑥(𝑥 = 𝑦𝜓) ↔ (∃𝑥 𝑥 = 𝑦𝜓))
74, 6mpbiran 952 . 2 (∃𝑥(𝑥 = 𝑦𝜓) ↔ 𝜓)
83, 7bitri 264 1 (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wex 1701  wnf 1705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  equsexhv  2146  sb56  2147  cleljustALT2  2185  sb10f  2455  dprd2d2  18375  poimirlem25  33101
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