Theorem nfmod 2564

Description: Bound-variable hypothesis builder for the at-most-one quantifier. Deduction version of nfmo 2565. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker nfmodv 2562 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 2561	1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535  Ⅎwnf 1781  ∃*wmo 2541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178  ax-13 2380

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-mo 2543

This theorem is referenced by:  nfmo  2565