Theorem nfralya 2504

Description: Not-free for restricted universal quantification where 𝑦 and 𝐴 are distinct. See nfralxy 2502 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)

Hypotheses

Ref	Expression
nfralya.1	⊢ Ⅎ𝑥𝐴
nfralya.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfralya	⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑

Distinct variable group:   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem nfralya

Step	Hyp	Ref	Expression
1		nftru 1453	. . 3 ⊢ Ⅎ𝑦⊤
2		nfralya.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
4		nfralya.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	4	a1i 9	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
6	1, 3, 5	nfraldya 2499	. 2 ⊢ (⊤ → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1351	1 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1343  Ⅎwnf 1447  Ⅎwnfc 2293  ∀wral 2442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447

This theorem is referenced by:  nfiinya  3889  nfsup  6948  caucvgsrlemgt1  7727  axpre-suploclemres  7833  supinfneg  9524  infsupneg  9525  ctiunctlemudc  12307  trirec0  13757