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Theorem bj-spimedv 33056
Description: Version of spimed 2417 with a dv condition, which does not require ax-13 2408. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-spimedv.1 (𝜒 → Ⅎ𝑥𝜑)
bj-spimedv.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
bj-spimedv (𝜒 → (𝜑 → ∃𝑥𝜓))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem bj-spimedv
StepHypRef Expression
1 bj-spimedv.1 . . 3 (𝜒 → Ⅎ𝑥𝜑)
21nf5rd 2220 . 2 (𝜒 → (𝜑 → ∀𝑥𝜑))
3 ax6ev 2059 . . . 4 𝑥 𝑥 = 𝑦
4 bj-spimedv.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4eximii 1912 . . 3 𝑥(𝜑𝜓)
6519.35i 1958 . 2 (∀𝑥𝜑 → ∃𝑥𝜓)
72, 6syl6 35 1 (𝜒 → (𝜑 → ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1629  wex 1852  wnf 1856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203
This theorem depends on definitions:  df-bi 197  df-ex 1853  df-nf 1858
This theorem is referenced by:  bj-spimev  33057
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