Theorem reuimrmo 3709

Description: Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2644. (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 3371	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜓)
2		rmoim 3704	. 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 3077  ∃!wreu 3366  ∃*wrmo 3367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1801  df-mo 2567  df-eu 2597  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369

This theorem is referenced by:  2reurmo  3723