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Theorem 19.23h 1509
Description: Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 1-Feb-2015.)
Hypothesis
Ref Expression
19.23h.1 (𝜓 → ∀𝑥𝜓)
Assertion
Ref Expression
19.23h (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Proof of Theorem 19.23h
StepHypRef Expression
1 19.23h.1 . . 3 (𝜓 → ∀𝑥𝜓)
21ax-gen 1460 . 2 𝑥(𝜓 → ∀𝑥𝜓)
3 19.23ht 1508 . 2 (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
42, 3ax-mp 5 1 (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wal 1362  wex 1503
This theorem was proved from axioms:  ax-mp 5  ax-gen 1460  ax-ie2 1505
This theorem is referenced by:  alnex  1510  19.8a  1601  exlimih  1604  exlimdh  1607  nf2  1679  equs5or  1841  19.23v  1894  pm11.53  1907
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