Theorem nfiotaw 6318

Description: Bound-variable hypothesis builder for the ℩ class. Version of nfiota 6320 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 23-Aug-2011.) (Revised by Gino Giotto, 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 1805	. . 3 ⊢ Ⅎ𝑦⊤
2		nfiotaw.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
4	1, 3	nfiotadw 6317	. 2 ⊢ (⊤ → Ⅎ𝑥(℩𝑦𝜑))
5	4	mptru 1544	1 ⊢ Ⅎ𝑥(℩𝑦𝜑)

Syntax hints:  ⊤wtru 1538  Ⅎwnf 1784  Ⅎwnfc 2961  ℩cio 6312

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-in 3943  df-ss 3952  df-sn 4568  df-uni 4839  df-iota 6314

This theorem is referenced by:  csbiota  6348  nffv  6680  nfsum1  15046  nfsumw  15047  nfcprod1  15264  nfcprod  15265  nfafv2  43466