Theorem moani 2550

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 2549	. 2 ⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ∃*𝑥(𝜓 ∧ 𝜑)

Syntax hints:   ∧ wa 395  ∃*wmo 2535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-mo 2537

This theorem is referenced by:  euxfr2w  3675  euxfr2  3677  rmoeq  3693  reuxfrd  3703  fvopab6  6969  mpofun  7476  1stconst  8036  2ndconst  8037  pwfir  9208  iunmapdisj  9921  axaddf  11043  axmulf  11044  joinval  18283  meetval  18297  reuxfrdf  32472