Theorem nfals 17311

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 17295	. 2 ⊢ (∀∃𝑦(𝜑 → 𝜓) ↔ (∀𝑦(𝜑 → 𝜓) ∧ ∃𝑦𝜑))
2		nfals.1	. . . . 5 ⊢ Ⅎ𝑥𝜑
3		nfals.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfim 1625	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
5	4	nfal 1629	. . 3 ⊢ Ⅎ𝑥∀𝑦(𝜑 → 𝜓)
6	2	nfex 1690	. . 3 ⊢ Ⅎ𝑥∃𝑦𝜑
7	5, 6	nfan 1618	. 2 ⊢ Ⅎ𝑥(∀𝑦(𝜑 → 𝜓) ∧ ∃𝑦𝜑)
8	1, 7	nfxfr 1527	1 ⊢ Ⅎ𝑥∀∃𝑦(𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104  ∀wal 1400  Ⅎwnf 1513  ∃wex 1545  ∀∃wals 17293

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-ial 1587  ax-i5r 1588

This proof depends on definitions:  df-bi 117  df-nf 1514  df-als 17295

This theorem is used by: (None)