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Theorem bj-eximALT 34361
 Description: Alternate proof of exim 1836 directly from alim 1813 by using df-ex 1783 (using duality of ∀ and ∃. (Contributed by BJ, 9-Dec-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
bj-eximALT (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem bj-eximALT
StepHypRef Expression
1 con3 156 . . . 4 ((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
21alimi 1814 . . 3 (∀𝑥(𝜑𝜓) → ∀𝑥𝜓 → ¬ 𝜑))
3 alim 1813 . . 3 (∀𝑥𝜓 → ¬ 𝜑) → (∀𝑥 ¬ 𝜓 → ∀𝑥 ¬ 𝜑))
4 con3 156 . . 3 ((∀𝑥 ¬ 𝜓 → ∀𝑥 ¬ 𝜑) → (¬ ∀𝑥 ¬ 𝜑 → ¬ ∀𝑥 ¬ 𝜓))
52, 3, 43syl 18 . 2 (∀𝑥(𝜑𝜓) → (¬ ∀𝑥 ¬ 𝜑 → ¬ ∀𝑥 ¬ 𝜓))
6 df-ex 1783 . 2 (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
7 df-ex 1783 . 2 (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
85, 6, 73imtr4g 300 1 (∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537  ∃wex 1782 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812 This theorem depends on definitions:  df-bi 210  df-ex 1783 This theorem is referenced by:  bj-aleximiALT  34362
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