Theorem nfiotaw 6500

Description: Bound-variable hypothesis builder for the ℩ class. Version of nfiota 6502 with a disjoint variable condition, which does not require ax-13 2406. (Contributed by NM, 23-Aug-2011.) Avoid ax-13 2406. (Revised by GG, 26-Jan-2024.)

Hypothesis

Ref	Expression
nfiotaw.1	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfiotaw	⊢ Ⅎ𝑥(℩𝑦𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem nfiotaw

Step	Hyp	Ref	Expression
1		nftru 1837	. . 3 ⊢ Ⅎ𝑦⊤
2		nfiotaw.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
4	1, 3	nfiotadw 6499	. 2 ⊢ (⊤ → Ⅎ𝑥(℩𝑦𝜑))
5	4	mptru 1577	1 ⊢ Ⅎ𝑥(℩𝑦𝜑)

This proof depends on syntax axioms:  ⊤wtru 1571  Ⅎwnf 1816  Ⅎwnfc 2912  ℩cio 6494

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-v 3459  df-ss 3923  df-sn 4592  df-uni 4875  df-iota 6496

This theorem is used by:  csbiota  6533  nffv  6895  nfsum1  15760  nfsum  15761  nfcprod1  15980  nfcprod  15981  nfafv2  47988