Theorem moim 2544

Description: The at-most-one quantifier reverses implication. (Contributed by NM, 22-Apr-1995.)

Assertion

Ref	Expression
moim	⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))

Proof of Theorem moim

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imim1 83	. . . 4 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦)))
2	1	al2imi 1817	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝑥 = 𝑦) → ∀𝑥(𝜑 → 𝑥 = 𝑦)))
3	2	eximdv 1919	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
4		dfmo 2540	. 2 ⊢ (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦))
5		dfmo 2540	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
6	3, 4, 5	3imtr4g 296	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1781  ∃*wmo 2537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-mo 2539

This theorem is referenced by:  moimi  2545  moimdv  2546  mobi  2547  euimmo  2616  moexexlem  2626  rmoim  3686  rmoimi2  3689  ssrmof  3989  disjss3  5084  funmo  6514  uptx  23590  taylf  26326