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Mirrors > Home > ILE Home > Th. List > 2fveq3 | GIF version |
Description: Equality theorem for nested function values. (Contributed by AV, 14-Aug-2022.) |
Ref | Expression |
---|---|
2fveq3 | ⊢ (𝐴 = 𝐵 → (𝐹‘(𝐺‘𝐴)) = (𝐹‘(𝐺‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 5534 | . 2 ⊢ (𝐴 = 𝐵 → (𝐺‘𝐴) = (𝐺‘𝐵)) | |
2 | 1 | fveq2d 5538 | 1 ⊢ (𝐴 = 𝐵 → (𝐹‘(𝐺‘𝐴)) = (𝐹‘(𝐺‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ‘cfv 5235 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rex 2474 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-iota 5196 df-fv 5243 |
This theorem is referenced by: difinfsnlem 7129 ctssdclemn0 7140 cc2 7297 seq3f1olemqsum 10533 seq3f1oleml 10536 seq3f1o 10537 seq3homo 10543 seq3coll 10857 fsumf1o 11433 iserabs 11518 explecnv 11548 cvgratnnlemnexp 11567 cvgratnnlemmn 11568 fprodf1o 11631 alginv 12082 algcvg 12083 algcvga 12086 ctiunctlemu1st 12488 ctiunctlemu2nd 12489 ctiunctlemudc 12491 ctiunctlemfo 12493 isunitd 13473 subctctexmid 15229 |
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