Theorem nfimd 1916

Description: If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 → 𝜒). Deduction form of nfim 1918. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) df-nf 1806 changed. (Revised by Wolf Lammen, 18-Sep-2021.) Eliminate curried form of nfimt 1917. (Revised by Wolf Lammen, 10-Jul-2022.)

Hypotheses

Ref	Expression
nfimd.1	⊢ (𝜑 → Ⅎ𝑥𝜓)
nfimd.2	⊢ (𝜑 → Ⅎ𝑥𝜒)

Assertion

Ref	Expression
nfimd	⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒))

Proof of Theorem nfimd

Step	Hyp	Ref	Expression
1		19.35 1899	. . . 4 ⊢ (∃𝑥(𝜓 → 𝜒) ↔ (∀𝑥𝜓 → ∃𝑥𝜒))
2	1	biimpi 218	. . 3 ⊢ (∃𝑥(𝜓 → 𝜒) → (∀𝑥𝜓 → ∃𝑥𝜒))
3		nfimd.1	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4	3	nfrd 1813	. . . 4 ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
5		nfimd.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜒)
6	5	nfrd 1813	. . . 4 ⊢ (𝜑 → (∃𝑥𝜒 → ∀𝑥𝜒))
7	4, 6	imim12d 81	. . 3 ⊢ (𝜑 → ((∀𝑥𝜓 → ∃𝑥𝜒) → (∃𝑥𝜓 → ∀𝑥𝜒)))
8		19.38 1861	. . 3 ⊢ ((∃𝑥𝜓 → ∀𝑥𝜒) → ∀𝑥(𝜓 → 𝜒))
9	2, 7, 8	syl56 36	. 2 ⊢ (𝜑 → (∃𝑥(𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒)))
10	9	nfd 1812	1 ⊢ (𝜑 → Ⅎ𝑥(𝜓 → 𝜒))

Syntax hints:   → wi 4  ∀wal 1560  ∃wex 1801  Ⅎwnf 1805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831

This theorem depends on definitions:  df-bi 209  df-ex 1802  df-nf 1806

This theorem is referenced by:  nfimt  1917  nfand  1919  nfbid  1924  nfim1  2236  hbimd  2334  dvelimhw  2378  dvelimf  2481  nfmod2  2587  nfmodv  2588  nfabdw  2947  nfraldw  3309  nfrald  3361  nfifd  4512  nfixpw  8900  nfixp  8901  axrepndlem1  10552  axrepndlem2  10553  axunndlem1  10555  axunnd  10556  axpowndlem2  10558  axpowndlem3  10559  axpowndlem4  10560  axregndlem2  10563  axregnd  10564  axinfndlem1  10565  axinfnd  10566  axacndlem4  10570  axacndlem5  10571  axacnd  10572  axpowg2  35447  axpowg3  35448  mh-setindnd  36902  bj-dvelimdv  37341  wl-mo2df  38078  wl-mo2t  38083  riotasv2d  39586  nfintd  50299