Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > nfsbc | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for class substitution. Usage of this theorem is discouraged because it depends on ax-13 2390. Use the weaker nfsbcw 3796 when possible. (Contributed by NM, 7-Sep-2014.) (Revised by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfsbc.1 | ⊢ Ⅎ𝑥𝐴 |
nfsbc.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfsbc | ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1805 | . . 3 ⊢ Ⅎ𝑦⊤ | |
2 | nfsbc.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
4 | nfsbc.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
6 | 1, 3, 5 | nfsbcd 3798 | . 2 ⊢ (⊤ → Ⅎ𝑥[𝐴 / 𝑦]𝜑) |
7 | 6 | mptru 1544 | 1 ⊢ Ⅎ𝑥[𝐴 / 𝑦]𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1538 Ⅎwnf 1784 Ⅎwnfc 2963 [wsbc 3774 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-sbc 3775 |
This theorem is referenced by: cbvralcsf 3927 ralrnmpt 6864 elovmporab1 7395 opreu2reuALT 30242 |
Copyright terms: Public domain | W3C validator |