Theorem moimd 29326

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 1855	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
3		moim 2519	. 2 ⊢ (∀𝑥(𝜓 → 𝜒) → (∃*𝑥𝜒 → ∃*𝑥𝜓))
4	2, 3	syl 17	1 ⊢ (𝜑 → (∃*𝑥𝜒 → ∃*𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1481  ∃*wmo 2471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-eu 2474  df-mo 2475

This theorem is referenced by: (None)