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Mirrors > Home > MPE Home > Th. List > nfral | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker nfralw 3138 when possible. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfral.1 | ⊢ Ⅎ𝑥𝐴 |
nfral.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfral | ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1811 | . . 3 ⊢ Ⅎ𝑦⊤ | |
2 | nfral.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
4 | nfral.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
6 | 1, 3, 5 | nfrald 3137 | . 2 ⊢ (⊤ → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑) |
7 | 6 | mptru 1549 | 1 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1543 Ⅎwnf 1790 Ⅎwnfc 2879 ∀wral 3053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-13 2372 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 |
This theorem is referenced by: nfra2 3142 nfiing 4914 opreu2reuALT 30399 eliuniincex 42197 cbvral2 44127 |
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