Theorem nfals 51327

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 51312	. 2 ⊢ (∀∃𝑦(𝜑 → 𝜓) ↔ (∀𝑦(𝜑 → 𝜓) ∧ ∃𝑦𝜑))
2		nfals.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
3		nfals.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfim 1931	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
5	4	nfal 2357	. . 3 ⊢ Ⅎ𝑥∀𝑦(𝜑 → 𝜓)
6	2	nfex 2358	. . 3 ⊢ Ⅎ𝑥∃𝑦𝜑
7	5, 6	nfan 1934	. 2 ⊢ Ⅎ𝑥(∀𝑦(𝜑 → 𝜓) ∧ ∃𝑦𝜑)
8	1, 7	nfxfr 1886	1 ⊢ Ⅎ𝑥∀∃𝑦(𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816  ∀∃wals 51310

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-11 2197  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-als 51312

This theorem is used by: (None)