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| Mirrors > Home > MPE Home > Th. List > imbrov2fvoveq | Structured version Visualization version 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 6906 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝐺‘𝑋) = (𝐺‘𝑌)) | |
| 3 | 2 | fvoveq1d 7453 | . . 3 ⊢ (𝑋 = 𝑌 → (𝐹‘((𝐺‘𝑋) · 𝑂)) = (𝐹‘((𝐺‘𝑌) · 𝑂))) |
| 4 | 3 | breq1d 5153 | . 2 ⊢ (𝑋 = 𝑌 → ((𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴 ↔ (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴)) |
| 5 | 1, 4 | imbi12d 344 | 1 ⊢ (𝑋 = 𝑌 → ((𝜑 → (𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴) ↔ (𝜓 → (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: rlim2 15532 rlimclim1 15581 rlimcn1 15624 climcn1 15628 caucvgrlem 15709 cncfco 24933 ftc1lem4 26080 ftc1lem6 26082 itg2gt0cn 37682 ftc1cnnclem 37698 ftc1cnnc 37699 idlimc 45641 limcperiod 45643 limclner 45666 cncfshift 45889 cncfperiod 45894 fperdvper 45934 |
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