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Theorem bj-19.41al 32306
Description: Special case of 19.41 2101 proved from Tarski, ax-10 2016 (modal5) and hba1 2148 (modal4). (Contributed by BJ, 29-Dec-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-19.41al (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))

Proof of Theorem bj-19.41al
StepHypRef Expression
1 19.40 1794 . . 3 (∃𝑥(𝜑 ∧ ∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝑥𝜓))
2 bj-modal5e 32305 . . . 4 (∃𝑥𝑥𝜓 → ∀𝑥𝜓)
32anim2i 592 . . 3 ((∃𝑥𝜑 ∧ ∃𝑥𝑥𝜓) → (∃𝑥𝜑 ∧ ∀𝑥𝜓))
41, 3syl 17 . 2 (∃𝑥(𝜑 ∧ ∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∀𝑥𝜓))
5 hba1 2148 . . . 4 (∀𝑥𝜓 → ∀𝑥𝑥𝜓)
65anim2i 592 . . 3 ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∀𝑥𝑥𝜓))
7 19.29r 1799 . . 3 ((∃𝑥𝜑 ∧ ∀𝑥𝑥𝜓) → ∃𝑥(𝜑 ∧ ∀𝑥𝜓))
86, 7syl 17 . 2 ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ ∀𝑥𝜓))
94, 8impbii 199 1 (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wal 1478  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707
This theorem is referenced by:  bj-equsexval  32307
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