Theorem nfals 51082

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 51067	. 2 ⊢ (∀∃𝑦(𝜑 → 𝜓) ↔ (∀𝑦(𝜑 → 𝜓) ∧ ∃𝑦𝜑))
2		nfals.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
3		nfals.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfim 1928	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
5	4	nfal 2353	. . 3 ⊢ Ⅎ𝑥∀𝑦(𝜑 → 𝜓)
6	2	nfex 2354	. . 3 ⊢ Ⅎ𝑥∃𝑦𝜑
7	5, 6	nfan 1931	. 2 ⊢ Ⅎ𝑥(∀𝑦(𝜑 → 𝜓) ∧ ∃𝑦𝜑)
8	1, 7	nfxfr 1885	1 ⊢ Ⅎ𝑥∀∃𝑦(𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567  ∃wex 1811  Ⅎwnf 1815  ∀∃wals 51065

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-als 51067

This theorem is used by: (None)