Theorem moimd 28710

Description: "At most one" is preserved through implication (notice wff reversal). (Contributed by Thierry Arnoux, 25-Feb-2017.)

Hypothesis

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

Assertion

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

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem moimd

Step	Hyp	Ref	Expression
1		moimd.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	alrimiv 1842	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
3		moim 2507	. 2 ⊢ (∀𝑥(𝜓 → 𝜒) → (∃*𝑥𝜒 → ∃*𝑥𝜓))
4	2, 3	syl 17	1 ⊢ (𝜑 → (∃*𝑥𝜒 → ∃*𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1473  ∃*wmo 2459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-eu 2462  df-mo 2463

This theorem is referenced by: (None)