Theorem nfrals 50869

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

Hypotheses

Ref	Expression
nfrals.1	⊢ Ⅎ𝑥𝐴
nfrals.2	⊢ Ⅎ𝑥𝜑
nfrals.3	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
nfrals	⊢ Ⅎ𝑥∀∃𝑦 ∈ 𝐴(𝜑 → 𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑥, 𝑦)   𝐴(𝑥, 𝑦)

Proof of Theorem nfrals

Step	Hyp	Ref	Expression
1		df-rals 50854	. 2 ⊢ (∀∃𝑦 ∈ 𝐴(𝜑 → 𝜓) ↔ (∀𝑦 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑦 ∈ 𝐴 𝜑))
2		nfrals.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfrals.2	. . . . 5 ⊢ Ⅎ𝑥𝜑
4		nfrals.3	. . . . 5 ⊢ Ⅎ𝑥𝜓
5	3, 4	nfim 1929	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
6	2, 5	nfralw 3310	. . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 (𝜑 → 𝜓)
7	2, 3	nfrexw 3311	. . 3 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑
8	6, 7	nfan 1932	. 2 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑦 ∈ 𝐴 𝜑)
9	1, 8	nfxfr 1886	1 ⊢ Ⅎ𝑥∀∃𝑦 ∈ 𝐴(𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  Ⅎwnf 1816  Ⅎwnfc 2908  ∀wral 3077  ∃wrex 3087  ∀∃wrals 50852

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-10 2178  ax-11 2194  ax-12 2213

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rals 50854

This theorem is used by: (None)