Theorem rmoan 2930

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 2088	. . 3 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) → ∃*𝑥(𝜓 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
2		an12 556	. . . 4 ⊢ ((𝜓 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ (𝜓 ∧ 𝜑)))
3	2	mobii 2056	. . 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 2456	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
6		df-rmo 2456	. 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 2020   ∈ wcel 2141  ∃*wrmo 2451

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-rmo 2456

This theorem is referenced by: (None)