Theorem rmoimia 3037

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 3036	. 2 ⊢ (∀x ∈ A (φ → ψ) → (∃*x ∈ A ψ → ∃*x ∈ A φ))
2		rmoimia.1	. 2 ⊢ (x ∈ A → (φ → ψ))
3	1, 2	mprg 2684	1 ⊢ (∃*x ∈ A ψ → ∃*x ∈ A φ)

Syntax hints:   → wi 4   ∈ wcel 1710  ∃*wrmo 2618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-ral 2620  df-rmo 2623

This theorem is referenced by: (None)