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Theorem r19.28m 3338
Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
Hypothesis
Ref Expression
r19.28m.1 𝑥𝜑
Assertion
Ref Expression
r19.28m (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem r19.28m
StepHypRef Expression
1 r19.28m.1 . . . 4 𝑥𝜑
21r19.3rm 3337 . . 3 (∃𝑥 𝑥𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
32anbi1d 446 . 2 (∃𝑥 𝑥𝐴 → ((𝜑 ∧ ∀𝑥𝐴 𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓)))
4 r19.26 2458 . 2 (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑥𝐴 𝜓))
53, 4syl6rbbr 192 1 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wnf 1365  wex 1397  wcel 1409  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-cleq 2049  df-clel 2052  df-ral 2328
This theorem is referenced by:  r19.28mv  3341  raaanlem  3353
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