Theorem bj-eximALT 34822

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(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem bj-eximALT

Step	Hyp	Ref	Expression
1		con3 153	. . . 4 ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
2	1	alimi 1814	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑥(¬ 𝜓 → ¬ 𝜑))
3		alim 1813	. . 3 ⊢ (∀𝑥(¬ 𝜓 → ¬ 𝜑) → (∀𝑥 ¬ 𝜓 → ∀𝑥 ¬ 𝜑))
4		con3 153	. . 3 ⊢ ((∀𝑥 ¬ 𝜓 → ∀𝑥 ¬ 𝜑) → (¬ ∀𝑥 ¬ 𝜑 → ¬ ∀𝑥 ¬ 𝜓))
5	2, 3, 4	3syl 18	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (¬ ∀𝑥 ¬ 𝜑 → ¬ ∀𝑥 ¬ 𝜓))
6		df-ex 1783	. 2 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
7		df-ex 1783	. 2 ⊢ (∃𝑥𝜓 ↔ ¬ ∀𝑥 ¬ 𝜓)
8	5, 6, 7	3imtr4g 296	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))

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 206  df-ex 1783

This theorem is referenced by:  bj-aleximiALT  34823