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Theorem 19.23vOLD 2086
Description: Obsolete version of 19.23v 2038 as of 15-Apr-2022. (Contributed by NM, 28-Jun-1998.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 11-Jan-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
19.23vOLD (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Distinct variable group:   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem 19.23vOLD
StepHypRef Expression
1 exim 1929 . . 3 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
2 19.9v 2080 . . 3 (∃𝑥𝜓𝜓)
31, 2syl6ib 243 . 2 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑𝜓))
4 ax-5 2006 . . . 4 (𝜓 → ∀𝑥𝜓)
54imim2i 16 . . 3 ((∃𝑥𝜑𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓))
6 19.38 1934 . . 3 ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
75, 6syl 17 . 2 ((∃𝑥𝜑𝜓) → ∀𝑥(𝜑𝜓))
83, 7impbii 201 1 (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1651  wex 1875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072
This theorem depends on definitions:  df-bi 199  df-ex 1876
This theorem is referenced by: (None)
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