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Theorem 19.36 2096
Description: Theorem 19.36 of [Margaris] p. 90. See 19.36v 1852 for a version requiring fewer axioms. (Contributed by NM, 24-Jun-1993.)
Hypothesis
Ref Expression
19.36.1 𝑥𝜓
Assertion
Ref Expression
19.36 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Proof of Theorem 19.36
StepHypRef Expression
1 19.35 1771 . 2 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
2 19.36.1 . . . 4 𝑥𝜓
3219.9 2023 . . 3 (∃𝑥𝜓𝜓)
43imbi2i 321 . 2 ((∀𝑥𝜑 → ∃𝑥𝜓) ↔ (∀𝑥𝜑𝜓))
51, 4bitri 259 1 (∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 191  wal 1466  wex 1692  wnf 1696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-12 1983
This theorem depends on definitions:  df-bi 192  df-ex 1693  df-nf 1697
This theorem is referenced by:  19.36i  2097  19.12vv  2124  spcimgft  3146  19.12b  30599
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