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Theorem bj-19.41al 36044
Description: Special case of 19.41 2220 proved from core axioms, ax-10 2129 (modal5), and hba1 2281 (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 1881 . . 3 (∃𝑥(𝜑 ∧ ∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝑥𝜓))
2 hbe1a 2132 . . . 4 (∃𝑥𝑥𝜓 → ∀𝑥𝜓)
32anim2i 616 . . 3 ((∃𝑥𝜑 ∧ ∃𝑥𝑥𝜓) → (∃𝑥𝜑 ∧ ∀𝑥𝜓))
41, 3syl 17 . 2 (∃𝑥(𝜑 ∧ ∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∀𝑥𝜓))
5 hba1 2281 . . . 4 (∀𝑥𝜓 → ∀𝑥𝑥𝜓)
65anim2i 616 . . 3 ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → (∃𝑥𝜑 ∧ ∀𝑥𝑥𝜓))
7 19.29r 1869 . . 3 ((∃𝑥𝜑 ∧ ∀𝑥𝑥𝜓) → ∃𝑥(𝜑 ∧ ∀𝑥𝜓))
86, 7syl 17 . 2 ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ ∀𝑥𝜓))
94, 8impbii 208 1 (∃𝑥(𝜑 ∧ ∀𝑥𝜓) ↔ (∃𝑥𝜑 ∧ ∀𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wal 1531  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-10 2129  ax-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778
This theorem is referenced by:  bj-equsexval  36045
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