Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > oveq | GIF version |
Description: Equality theorem for operation value. (Contributed by NM, 28-Feb-1995.) |
Ref | Expression |
---|---|
oveq | ⊢ (𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 5413 | . 2 ⊢ (𝐹 = 𝐺 → (𝐹‘〈𝐴, 𝐵〉) = (𝐺‘〈𝐴, 𝐵〉)) | |
2 | df-ov 5770 | . 2 ⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) | |
3 | df-ov 5770 | . 2 ⊢ (𝐴𝐺𝐵) = (𝐺‘〈𝐴, 𝐵〉) | |
4 | 1, 2, 3 | 3eqtr4g 2195 | 1 ⊢ (𝐹 = 𝐺 → (𝐴𝐹𝐵) = (𝐴𝐺𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 〈cop 3525 ‘cfv 5118 (class class class)co 5767 |
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 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-uni 3732 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 |
This theorem is referenced by: oveqi 5780 oveqd 5784 ovmpodf 5895 ovmpodv2 5897 mapxpen 6735 ispsmet 12481 ismet 12502 isxmet 12503 |
Copyright terms: Public domain | W3C validator |