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 2401. (Contributed by NM, 23-Aug-2011.) Avoid ax-13 2401. (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 1836	. . 3 ⊢ Ⅎ𝑦⊤
2		nfiotaw.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
4	1, 3	nfiotadw 6499	. 2 ⊢ (⊤ → Ⅎ𝑥(℩𝑦𝜑))
5	4	mptru 1576	1 ⊢ Ⅎ𝑥(℩𝑦𝜑)

This proof depends on syntax axioms:  ⊤wtru 1570  Ⅎwnf 1815  Ⅎwnfc 2907  ℩cio 6494

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-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-v 3452  df-ss 3897  df-sn 4585  df-uni 4867  df-iota 6496

This theorem is used by:  csbiota  6534  nffv  6898  nfsum1  15864  nfsum  15865  nfcprod1  16084  nfcprod  16085  nfafv2  48395