Theorem moimdv 2563

Description: The at-most-one quantifier reverses implication, deduction form. (Contributed by Thierry Arnoux, 25-Feb-2017.)

Hypothesis

Ref	Expression
moimdv.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
moimdv	⊢ (𝜑 → (∃*𝑥𝜒 → ∃*𝑥𝜓))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem moimdv

Step	Hyp	Ref	Expression
1		moimdv.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	alrimiv 1928	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
3		moim 2561	. 2 ⊢ (∀𝑥(𝜓 → 𝜒) → (∃*𝑥𝜒 → ∃*𝑥𝜓))
4	2, 3	syl 17	1 ⊢ (𝜑 → (∃*𝑥𝜒 → ∃*𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1536  ∃*wmo 2555

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 1911

This theorem depends on definitions:  df-bi 210  df-ex 1782  df-mo 2557

This theorem is referenced by:  disjss1  5007  brdom6disj  10005  funressnfv  44046  funressnvmo  44048