Theorem reuimrmo 3703

Description: Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2641. (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 3368	. 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜓)
2		rmoim 3698	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑))
3	1, 2	syl5 35	1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃!𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑))

This proof depends on syntax axioms:   → wi 4  ∀wral 3076  ∃!wreu 3363  ∃*wrmo 3364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2564  df-eu 2594  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366

This theorem is used by:  2reurmo  3717