Theorem nfiotaw 6476

Description: Bound-variable hypothesis builder for the ℩ class. Version of nfiota 6478 with a disjoint variable condition, which does not require ax-13 2402. (Contributed by NM, 23-Aug-2011.) Avoid ax-13 2402. (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 1823	. . 3 ⊢ Ⅎ𝑦⊤
2		nfiotaw.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
4	1, 3	nfiotadw 6475	. 2 ⊢ (⊤ → Ⅎ𝑥(℩𝑦𝜑))
5	4	mptru 1566	1 ⊢ Ⅎ𝑥(℩𝑦𝜑)

Syntax hints:  ⊤wtru 1560  Ⅎwnf 1802  Ⅎwnfc 2908  ℩cio 6470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-v 3455  df-ss 3919  df-sn 4580  df-uni 4863  df-iota 6472

This theorem is referenced by:  csbiota  6509  nffv  6872  nfsum1  15708  nfsum  15709  nfcprod1  15929  nfcprod  15930  nfafv2  47773