Theorem nfmod 2561

Description: Bound-variable hypothesis builder for the at-most-one quantifier. Deduction version of nfmo 2562. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfmodv 2559 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 480	. 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
4	1, 3	nfmod2 2558	1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537  Ⅎwnf 1787  ∃*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  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-mo 2540

This theorem is referenced by:  nfmo  2562  wl-mo3t  35658