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Mirrors > Home > ILE Home > Th. List > nfralw | GIF version |
Description: Bound-variable hypothesis builder for restricted quantification. See nfralya 2497 for a version with 𝑦 and 𝐴 distinct instead of 𝑥 and 𝑦. (Contributed by NM, 1-Sep-1999.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
nfralw.1 | ⊢ Ⅎ𝑥𝐴 |
nfralw.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfralw | ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1446 | . . 3 ⊢ Ⅎ𝑦⊤ | |
2 | nfralw.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | a1i 9 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
4 | nfralw.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | a1i 9 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
6 | 1, 3, 5 | nfraldw 2489 | . 2 ⊢ (⊤ → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑) |
7 | 6 | mptru 1344 | 1 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff set class |
Syntax hints: ⊤wtru 1336 Ⅎwnf 1440 Ⅎwnfc 2286 ∀wral 2435 |
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-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-4 1490 ax-17 1506 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 |
This theorem is referenced by: fprod2dlemstep 11523 fprodcom2fi 11527 |
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