Theorem moimdv 2576

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 1960	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
3		moim 2574	. 2 ⊢ (∀𝑥(𝜓 → 𝜒) → (∃*𝑥𝜒 → ∃*𝑥𝜓))
4	2, 3	syl 18	1 ⊢ (𝜑 → (∃*𝑥𝜒 → ∃*𝑥𝜓))

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃*wmo 2567

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 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2569

This theorem is used by:  disjss1  5084  brdom6disj  10532  funressnfv  47857  funressnvmo  47859