Theorem nfrals 17053

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 17037	. 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 1625	. . . 4 ⊢ Ⅎ𝑥(𝜑 → 𝜓)
6	2, 5	nfralw 2587	. . 3 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 (𝜑 → 𝜓)
7	2, 3	nfrexw 2589	. . 3 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑
8	6, 7	nfan 1618	. 2 ⊢ Ⅎ𝑥(∀𝑦 ∈ 𝐴 (𝜑 → 𝜓) ∧ ∃𝑦 ∈ 𝐴 𝜑)
9	1, 8	nfxfr 1527	1 ⊢ Ⅎ𝑥∀∃𝑦 ∈ 𝐴(𝜑 → 𝜓)

Syntax hints:   → wi 4   ∧ wa 104  Ⅎwnf 1513  Ⅎwnfc 2379  ∀wral 2528  ∃wrex 2529  ∀∃wrals 17035

This theorem was proved from 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-17 1579  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rals 17037

This theorem is referenced by: (None)