Theorem rmoimi 42116

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 3621	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2106  ∃*wrmo 3092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-mo 2550  df-ral 3094  df-rmo 3097

This theorem is referenced by:  2rexreu  42128