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Theorem r19.23 2424
 Description: Theorem 19.23 of [Margaris] p. 90 with restricted quantifiers. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 8-Oct-2016.)
Hypothesis
Ref Expression
r19.23.1 𝑥𝜓
Assertion
Ref Expression
r19.23 (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))

Proof of Theorem r19.23
StepHypRef Expression
1 r19.23.1 . 2 𝑥𝜓
2 r19.23t 2423 . 2 (Ⅎ𝑥𝜓 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
31, 2ax-mp 7 1 (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98  Ⅎwnf 1349  ∀wral 2306  ∃wrex 2307 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311  df-rex 2312 This theorem is referenced by:  r19.23v  2425  rexlimi  2426  rexlimd  2430
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