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Theorem 19.23h 1432
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 1383 . 2 𝑥(𝜓 → ∀𝑥𝜓)
3 19.23ht 1431 . 2 (∀𝑥(𝜓 → ∀𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓)))
42, 3ax-mp 7 1 (∀𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287  wex 1426
This theorem was proved from axioms:  ax-mp 7  ax-gen 1383  ax-ie2 1428
This theorem is referenced by:  alnex  1433  19.8a  1527  exlimih  1529  exlimdh  1532  nf2  1603  equs5or  1758  19.23v  1811  pm11.53  1823
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