Theorem rmoan 2926

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 2083	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃*𝑥(𝜓 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
2		an12 551	. . . 4 ⊢ ((𝜓 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑)))
3	2	mobii 2051	. . 3 ⊢ (∃*𝑥(𝜓 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑)))
4	1, 3	sylib 121	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑)))
5		df-rmo 2452	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6		df-rmo 2452	. 2 ⊢ (∃*𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑)))
7	4, 5, 6	3imtr4i 200	1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 → ∃*𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑))

Syntax hints:   → wi 4   ∧ wa 103  ∃*wmo 2015   ∈ wcel 2136  ∃*wrmo 2447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-rmo 2452

This theorem is referenced by: (None)