Theorem moani 2553

Description: "At most one" is still true when a conjunct is added. (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 2552	. 2 ⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ∃*𝑥(𝜓 ∧ 𝜑)

Syntax hints:   ∧ wa 395  ∃*wmo 2538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1784  df-mo 2540

This theorem is referenced by:  euxfr2w  3650  euxfr2  3652  rmoeq  3668  reuxfrd  3678  fvopab6  6890  mpofun  7376  1stconst  7911  2ndconst  7912  pwfir  8921  iunmapdisj  9710  axaddf  10832  axmulf  10833  joinval  18010  meetval  18024  reuxfrdf  30740