Theorem rmoan 3547

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 2662	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃*𝑥(𝜓 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
2		an12 873	. . . 4 ⊢ ((𝜓 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑)))
3	2	mobii 2630	. . 3 ⊢ (∃*𝑥(𝜓 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑)))
4	1, 3	sylib 208	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑)))
5		df-rmo 3058	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6		df-rmo 3058	. 2 ⊢ (∃*𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑)))
7	4, 5, 6	3imtr4i 281	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 2139  ∃*wmo 2608  ∃*wrmo 3053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-10 2168  ax-12 2196

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-eu 2611  df-mo 2612  df-rmo 3058

This theorem is referenced by:  reuxfr2d  5040  reuxfr3d  29658