Theorem moan 2582

Description: "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)

Assertion

Ref	Expression
moan	⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑))

Proof of Theorem moan

Step	Hyp	Ref	Expression
1		simpr 490	. 2 ⊢ ((𝜓 ∧ 𝜑) → 𝜑)
2	1	moimi 2575	1 ⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∃*wmo 2567

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 2569

This theorem is used by:  moani  2583  mooran1  2585  moanimlem  2648  mormo  3376  rmoan  3704