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Mirrors > Home > ILE Home > Th. List > imbrov2fvoveq | GIF version |
Description: Equality theorem for nested function and operation value in an implication for a binary relation. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 17-Aug-2022.) |
Ref | Expression |
---|---|
imbrov2fvoveq.1 | ⊢ (𝑋 = 𝑌 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
imbrov2fvoveq | ⊢ (𝑋 = 𝑌 → ((𝜑 → (𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴) ↔ (𝜓 → (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imbrov2fvoveq.1 | . 2 ⊢ (𝑋 = 𝑌 → (𝜑 ↔ 𝜓)) | |
2 | fveq2 5494 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝐺‘𝑋) = (𝐺‘𝑌)) | |
3 | 2 | fvoveq1d 5872 | . . 3 ⊢ (𝑋 = 𝑌 → (𝐹‘((𝐺‘𝑋) · 𝑂)) = (𝐹‘((𝐺‘𝑌) · 𝑂))) |
4 | 3 | breq1d 3997 | . 2 ⊢ (𝑋 = 𝑌 → ((𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴 ↔ (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴)) |
5 | 1, 4 | imbi12d 233 | 1 ⊢ (𝑋 = 𝑌 → ((𝜑 → (𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴) ↔ (𝜓 → (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1348 class class class wbr 3987 ‘cfv 5196 (class class class)co 5850 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-rex 2454 df-v 2732 df-un 3125 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-iota 5158 df-fv 5204 df-ov 5853 |
This theorem is referenced by: cncfco 13293 mulcncflem 13305 ivthinclemlopn 13329 ivthinclemuopn 13331 limcimolemlt 13348 eflt 13411 |
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