Theorem moani 2584

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 2583	. 2 ⊢ (∃*𝑥𝜑 → ∃*𝑥(𝜓 ∧ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ∃*𝑥(𝜓 ∧ 𝜑)

This proof depends on syntax axioms:   ∧ wa 401  ∃*wmo 2568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-mo 2570

This theorem is used by:  euxfr2w  3686  euxfr2  3688  rmoeq  3704  reuxfrd  3714  fvopab6  7031  mpofun  7547  1stconst  8104  2ndconst  8105  pwfir  9286  iunmapdisj  10026  axaddf  11148  axmulf  11149  joinval  18456  meetval  18470  reuxfrdf  32874