Theorem reuimrmo 3680

Description: Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2618. (Contributed by Alexander van der Vekens, 25-Jun-2017.)

Assertion

Ref	Expression
reuimrmo	⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃!𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑))

Proof of Theorem reuimrmo

Step	Hyp	Ref	Expression
1		reurmo 3364	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜓)
2		rmoim 3675	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑))
3	1, 2	syl5 34	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃!𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑))

Syntax hints:   → wi 4  ∀wral 3064  ∃!wreu 3066  ∃*wrmo 3067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-mo 2540  df-eu 2569  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072

This theorem is referenced by:  2reurmo  3694