Theorem nfald 2321

Description: Deduction form of nfal 2316. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 16-Oct-2021.)

Hypotheses

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

Assertion

Ref	Expression
nfald	⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)

Proof of Theorem nfald

Step	Hyp	Ref	Expression
1		19.12 2320	. . 3 ⊢ (∃𝑥∀𝑦𝜓 → ∀𝑦∃𝑥𝜓)
2		nfald.1	. . . 4 ⊢ Ⅎ𝑦𝜑
3		nfald.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4	3	nfrd 1792	. . . 4 ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
5	2, 4	alimd 2204	. . 3 ⊢ (𝜑 → (∀𝑦∃𝑥𝜓 → ∀𝑦∀𝑥𝜓))
6		ax-11 2153	. . 3 ⊢ (∀𝑦∀𝑥𝜓 → ∀𝑥∀𝑦𝜓)
7	1, 5, 6	syl56 36	. 2 ⊢ (𝜑 → (∃𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜓))
8	7	nfd 1791	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1538  ∃wex 1780  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-11 2153  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-or 845  df-ex 1781  df-nf 1785

This theorem is referenced by:  nfexd  2322  dvelimhw  2342  nfald2  2444  nfmodv  2558  nfeqd  2915  nfabdw  2928  nfraldw  3289  nfraldwOLD  3299  nfiotadw  6420  nfixpw  8752  axrepndlem1  10421  axrepndlem2  10422  axunnd  10425  axpowndlem2  10427  axpowndlem4  10429  axregndlem2  10432  axinfndlem1  10434  axinfnd  10435  axacndlem4  10439  axacndlem5  10440  axacnd  10441  bj-dvelimdv  35092  wl-mo2df  35781  wl-eudf  35783  wl-mo2t  35786  nfintd  46631