Theorem nfmod 2586

Description: Bound-variable hypothesis builder for the at-most-one quantifier. Deduction version of nfmo 2587. Usage of this theorem is discouraged because it depends on ax-13 2401. Use the weaker nfmodv 2584 when possible. (Contributed by Mario Carneiro, 14-Nov-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfmod.1	⊢ Ⅎ𝑦𝜑
nfmod.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfmod	⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Proof of Theorem nfmod

Step	Hyp	Ref	Expression
1		nfmod.1	. 2 ⊢ Ⅎ𝑦𝜑
2		nfmod.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3	2	adantr 486	. 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
4	1, 3	nfmod2 2583	1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1567  Ⅎwnf 1815  ∃*wmo 2562

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212  ax-13 2401

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-mo 2564

This theorem is used by:  nfmo  2587