Theorem nfals 50581

Description: Bound-variable hypothesis builder for "all some". (Contributed by David A. Wheeler, 12-Jul-2026.)

Hypotheses

Ref	Expression
nfals.1	⊢ Ⅎ𝑥𝜑
nfals.2	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
nfals	⊢ Ⅎ𝑥∀∃𝑦(𝜑 → 𝜓)

Proof of Theorem nfals

Step	Hyp	Ref	Expression
1		df-als 50566	. 2 ⊢ (∀∃𝑦(𝜑 → 𝜓) ↔ (∀𝑦(𝜑 → 𝜓) ∧ ∃𝑦𝜑))
2		nfals.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
3		nfals.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfim 1926	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
5	4	nfal 2356	. . 3 ⊢ Ⅎ𝑥∀𝑦(𝜑 → 𝜓)
6	2	nfex 2357	. . 3 ⊢ Ⅎ𝑥∃𝑦𝜑
7	5, 6	nfan 1929	. 2 ⊢ Ⅎ𝑥(∀𝑦(𝜑 → 𝜓) ∧ ∃𝑦𝜑)
8	1, 7	nfxfr 1883	1 ⊢ Ⅎ𝑥∀∃𝑦(𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1568  ∃wex 1809  Ⅎwnf 1813  ∀∃wals 50564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-als 50566

This theorem is referenced by: (None)