Theorem moani 2557

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 2556	. 2 ⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ∃*𝑥(𝜓 ∧ 𝜑)

Syntax hints:   ∧ wa 396  ∃*wmo 2541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-mo 2543

This theorem is referenced by:  euxfr2w  3668  euxfr2  3670  rmoeq  3686  reuxfrd  3696  fvopab6  6977  mpofun  7487  1stconst  8046  2ndconst  8047  pwfir  9224  iunmapdisj  9943  axaddf  11066  axmulf  11067  joinval  18339  meetval  18353  reuxfrdf  32585