Theorem rmoimi 3738

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

Hypothesis

Ref	Expression
rmoimi.1	⊢ (𝜑 → 𝜓)

Assertion

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

Proof of Theorem rmoimi

Step	Hyp	Ref	Expression
1		rmoimi.1	. . 3 ⊢ (𝜑 → 𝜓)
2	1	a1i 11	. 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓))
3	2	rmoimia 3737	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2105  ∃*wrmo 3374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781  df-mo 2533  df-ral 3061  df-rmo 3375

This theorem is referenced by:  2rexreu  3758  2sqreunnlem1  27203  disjin  32099  disjin2  32100  addinvcom  41619