Theorem reuimrmo 3742

Description: Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2611. (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 3378	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜓)
2		rmoim 3737	. 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 3060  ∃!wreu 3373  ∃*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-eu 2562  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376

This theorem is referenced by:  2reurmo  3756