Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > csbfv12g | GIF version |
Description: Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.) |
Ref | Expression |
---|---|
csbfv12g | ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbiotag 5116 | . . 3 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(℩𝑦𝐵𝐹𝑦) = (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦)) | |
2 | sbcbrg 3982 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → ([𝐴 / 𝑥]𝐵𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦)) | |
3 | csbconstg 3016 | . . . . . 6 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦) | |
4 | 3 | breq2d 3941 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → (⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦)) |
5 | 2, 4 | bitrd 187 | . . . 4 ⊢ (𝐴 ∈ 𝐶 → ([𝐴 / 𝑥]𝐵𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦)) |
6 | 5 | iotabidv 5109 | . . 3 ⊢ (𝐴 ∈ 𝐶 → (℩𝑦[𝐴 / 𝑥]𝐵𝐹𝑦) = (℩𝑦⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦)) |
7 | 1, 6 | eqtrd 2172 | . 2 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(℩𝑦𝐵𝐹𝑦) = (℩𝑦⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦)) |
8 | df-fv 5131 | . . 3 ⊢ (𝐹‘𝐵) = (℩𝑦𝐵𝐹𝑦) | |
9 | 8 | csbeq2i 3029 | . 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = ⦋𝐴 / 𝑥⦌(℩𝑦𝐵𝐹𝑦) |
10 | df-fv 5131 | . 2 ⊢ (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = (℩𝑦⦋𝐴 / 𝑥⦌𝐵⦋𝐴 / 𝑥⦌𝐹𝑦) | |
11 | 7, 9, 10 | 3eqtr4g 2197 | 1 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 [wsbc 2909 ⦋csb 3003 class class class wbr 3929 ℩cio 5086 ‘cfv 5123 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-fv 5131 |
This theorem is referenced by: csbfv2g 5458 |
Copyright terms: Public domain | W3C validator |