Theorem nfiotaw 6449

Description: Bound-variable hypothesis builder for the ℩ class. Version of nfiota 6451 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by NM, 23-Aug-2011.) Avoid ax-13 2374. (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 1805	. . 3 ⊢ Ⅎ𝑦⊤
2		nfiotaw.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
4	1, 3	nfiotadw 6448	. 2 ⊢ (⊤ → Ⅎ𝑥(℩𝑦𝜑))
5	4	mptru 1548	1 ⊢ Ⅎ𝑥(℩𝑦𝜑)

Syntax hints:  ⊤wtru 1542  Ⅎwnf 1784  Ⅎwnfc 2880  ℩cio 6443

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-rex 3058  df-v 3439  df-ss 3915  df-sn 4578  df-uni 4861  df-iota 6445

This theorem is referenced by:  csbiota  6482  nffv  6841  nfsum1  15604  nfsum  15605  nfcprod1  15822  nfcprod  15823  nfafv2  47380