Theorem moim 2572

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 84	. . . 4 ⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝑥 = 𝑦) → (𝜑 → 𝑥 = 𝑦)))
2	1	al2imi 1848	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝑥 = 𝑦) → ∀𝑥(𝜑 → 𝑥 = 𝑦)))
3	2	eximdv 1952	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦) → ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦)))
4		dfmo 2568	. 2 ⊢ (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦))
5		dfmo 2568	. 2 ⊢ (∃*𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦))
6	3, 4, 5	3imtr4g 299	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃*𝑥𝜓 → ∃*𝑥𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812  ∃*wmo 2565

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2567

This theorem is used by:  moimi  2573  moimdv  2574  mobi  2575  euimmo  2644  moexexlem  2654  rmoim  3701  rmoimi2  3704  ssrmof  3981  disjss3  5105  funmo  6574  uptx  23974  taylf  26718