![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nfcsb | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for substitution into a class. Usage of this theorem is discouraged because it depends on ax-13 2367. Use the weaker nfcsbw 3919 when possible. (Contributed by Mario Carneiro, 12-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfcsb.1 | ⊢ Ⅎ𝑥𝐴 |
nfcsb.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfcsb | ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nftru 1799 | . . 3 ⊢ Ⅎ𝑦⊤ | |
2 | nfcsb.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
4 | nfcsb.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐵) |
6 | 1, 3, 5 | nfcsbd 3918 | . 2 ⊢ (⊤ → Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵) |
7 | 6 | mptru 1541 | 1 ⊢ Ⅎ𝑥⦋𝐴 / 𝑦⦌𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1535 Ⅎwnfc 2879 ⦋csb 3892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-13 2367 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-sbc 3777 df-csb 3893 |
This theorem is referenced by: cbvralcsf 3937 cbvreucsf 3939 cbvrabcsf 3940 elfvmptrab1 7033 elovmporab1 7669 |
Copyright terms: Public domain | W3C validator |