Theorem bj-nfald 34451

Description: Variant of nfald 2346. (Contributed by BJ, 25-Dec-2023.)

Hypotheses

Ref	Expression
bj-nfald.1	⊢ (𝜑 → ∀𝑦𝜑)
bj-nfald.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
bj-nfald	⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)

Proof of Theorem bj-nfald

Step	Hyp	Ref	Expression
1		19.12 2345	. . 3 ⊢ (∃𝑥∀𝑦𝜓 → ∀𝑦∃𝑥𝜓)
2		bj-nfald.1	. . . 4 ⊢ (𝜑 → ∀𝑦𝜑)
3		bj-nfald.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
4	3	nfrd 1791	. . . 4 ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
5	2, 4	alimdh 1817	. . 3 ⊢ (𝜑 → (∀𝑦∃𝑥𝜓 → ∀𝑦∀𝑥𝜓))
6		ax-11 2160	. . 3 ⊢ (∀𝑦∀𝑥𝜓 → ∀𝑥∀𝑦𝜓)
7	1, 5, 6	syl56 36	. 2 ⊢ (𝜑 → (∃𝑥∀𝑦𝜓 → ∀𝑥∀𝑦𝜓))
8	7	nfd 1790	1 ⊢ (𝜑 → Ⅎ𝑥∀𝑦𝜓)

Syntax hints:   → wi 4  ∀wal 1534  ∃wex 1779  Ⅎwnf 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-10 2144  ax-11 2160  ax-12 2176

This theorem depends on definitions:  df-bi 209  df-or 844  df-ex 1780  df-nf 1784

This theorem is referenced by:  bj-nfexd  34452