Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > csbie | Structured version Visualization version GIF version |
Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by AV, 2-Dec-2019.) |
Ref | Expression |
---|---|
csbie.1 | ⊢ 𝐴 ∈ V |
csbie.2 | ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
csbie | ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbie.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | nfcv 2955 | . 2 ⊢ Ⅎ𝑥𝐶 | |
3 | csbie.2 | . 2 ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) | |
4 | 1, 2, 3 | csbief 3862 | 1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⦋csb 3828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-sbc 3721 df-csb 3829 |
This theorem is referenced by: pofun 5455 eqerlem 8306 mptnn0fsuppd 13361 fsum 15069 fsumcnv 15120 fsumshftm 15128 fsum0diag2 15130 fprod 15287 fprodcnv 15329 bpolyval 15395 ruclem1 15576 odfval 18652 odval 18654 psrass1lem 20615 selvval 20790 mamufval 20992 pm2mpval 21400 isibl 24369 dfitg 24373 dvfsumlem2 24630 fsumdvdsmul 25780 disjxpin 30351 poimirlem1 35058 poimirlem5 35062 poimirlem15 35072 poimirlem16 35073 poimirlem17 35074 poimirlem19 35076 poimirlem20 35077 poimirlem22 35079 poimirlem24 35081 poimirlem28 35085 fphpd 39757 monotuz 39882 oddcomabszz 39885 fnwe2val 39993 fnwe2lem1 39994 |
Copyright terms: Public domain | W3C validator |