Theorem rmoan 3674

Description: Restricted "at most one" still holds when a conjunct is added. (Contributed by NM, 16-Jun-2017.)

Assertion

Ref	Expression
rmoan	⊢ (∃*𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑))

Proof of Theorem rmoan

Step	Hyp	Ref	Expression
1		moan 2552	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃*𝑥(𝜓 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
2		an12 642	. . . 4 ⊢ ((𝜓 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑)))
3	2	mobii 2548	. . 3 ⊢ (∃*𝑥(𝜓 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑)))
4	1, 3	sylib 217	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑)))
5		df-rmo 3071	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6		df-rmo 3071	. 2 ⊢ (∃*𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑)))
7	4, 5, 6	3imtr4i 292	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  ∃*wmo 2538  ∃*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

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-mo 2540  df-rmo 3071

This theorem is referenced by:  reuxfrd  3683  reuxfrdf  30839