Theorem moani 2545

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 2544	. 2 ⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ∃*𝑥(𝜓 ∧ 𝜑)

Syntax hints:   ∧ wa 394  ∃*wmo 2530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1780  df-mo 2532

This theorem is referenced by:  euxfr2w  3717  euxfr2  3719  rmoeq  3735  reuxfrd  3745  fvopab6  7032  mpofun  7536  1stconst  8090  2ndconst  8091  pwfir  9180  iunmapdisj  10022  axaddf  11144  axmulf  11145  joinval  18336  meetval  18350  reuxfrdf  31996