Theorem rmoimia 3729

Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Hypothesis

Ref	Expression
rmoimia.1	⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))

Assertion

Ref	Expression
rmoimia	⊢ (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)

Proof of Theorem rmoimia

Step	Hyp	Ref	Expression
1		rmoim 3728	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑))
2		rmoimia.1	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	1, 2	mprg 3056	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2107  ∃*wrmo 3362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-mo 2538  df-ral 3051  df-rmo 3363

This theorem is referenced by:  rmoimi  3730  2reu1  3877