Theorem moani 2581

Description: "At most one" is still true when a conjunct is added, inference form. (Contributed by NM, 9-Mar-1995.)

Hypothesis

Ref	Expression
moani.1	⊢ ∃*𝑥𝜑

Assertion

Ref	Expression
moani	⊢ ∃*𝑥(𝜓 ∧ 𝜑)

Proof of Theorem moani

Step	Hyp	Ref	Expression
1		moani.1	. 2 ⊢ ∃*𝑥𝜑
2		moan 2580	. 2 ⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ∃*𝑥(𝜓 ∧ 𝜑)

Syntax hints:   ∧ wa 400  ∃*wmo 2565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-mo 2567

This theorem is referenced by:  euxfr2w  3684  euxfr2  3686  rmoeq  3702  reuxfrd  3712  fvopab6  7026  mpofun  7536  1stconst  8096  2ndconst  8097  pwfir  9277  iunmapdisj  10008  axaddf  11131  axmulf  11132  joinval  18432  meetval  18446  reuxfrdf  32818