Theorem nfexd 2337

Description: If 𝑥 is not free in 𝜓, it is not free in ∃𝑦𝜓. (Contributed by Mario Carneiro, 24-Sep-2016.)

Hypotheses

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

Assertion

Ref	Expression
nfexd	⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)

Proof of Theorem nfexd

Step	Hyp	Ref	Expression
1		df-ex 1782	. 2 ⊢ (∃𝑦𝜓 ↔ ¬ ∀𝑦 ¬ 𝜓)
2		nfald.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfald.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4	3	nfnd 1859	. . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
5	2, 4	nfald 2336	. . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ¬ 𝜓)
6	5	nfnd 1859	. 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ¬ 𝜓)
7	1, 6	nfxfrd 1855	1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1536  ∃wex 1781  Ⅎwnf 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-or 845  df-ex 1782  df-nf 1786

This theorem is referenced by:  nfmod2  2617  nfmodv  2618  nfeudw  2652  nfeld  2966  axrepndlem1  10003  axrepndlem2  10004  axunndlem1  10006  axunnd  10007  axpowndlem2  10009  axpowndlem3  10010  axpowndlem4  10011  axregndlem2  10014  axinfndlem1  10016  axinfnd  10017  axacndlem4  10021  axacndlem5  10022  axacnd  10023  19.9d2rf  30242  hbexg  41262