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Theorem r19.23 2598
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 2597 . 2 (Ⅎ𝑥𝜓 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
31, 2ax-mp 5 1 (∀𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  wnf 1471  wral 2468  wrex 2469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-nf 1472  df-ral 2473  df-rex 2474
This theorem is referenced by:  r19.23v  2599  rexlimi  2600  rexlimd  2604
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