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| Mirrors > Home > MPE Home > Th. List > Mathboxes > afveq12d | Structured version Visualization version GIF version | ||
| Description: Equality deduction for function value, analogous to fveq12d 6824. (Contributed by Alexander van der Vekens, 26-May-2017.) |
| Ref | Expression |
|---|---|
| afveq12d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
| afveq12d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| afveq12d | ⊢ (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | afveq12d.1 | . . . 4 ⊢ (𝜑 → 𝐹 = 𝐺) | |
| 2 | afveq12d.2 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 3 | 1, 2 | dfateq12d 47136 | . . 3 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) |
| 4 | 1, 2 | fveq12d 6824 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐵)) |
| 5 | 3, 4 | ifbieq1d 4498 | . 2 ⊢ (𝜑 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = if(𝐺 defAt 𝐵, (𝐺‘𝐵), V)) |
| 6 | dfafv2 47142 | . 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
| 7 | dfafv2 47142 | . 2 ⊢ (𝐺'''𝐵) = if(𝐺 defAt 𝐵, (𝐺‘𝐵), V) | |
| 8 | 5, 6, 7 | 3eqtr4g 2790 | 1 ⊢ (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1541 Vcvv 3434 ifcif 4473 ‘cfv 6477 defAt wdfat 47126 '''cafv 47127 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-sep 5232 ax-nul 5242 ax-pr 5368 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4282 df-if 4474 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-int 4896 df-br 5090 df-opab 5152 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-res 5626 df-iota 6433 df-fun 6479 df-fv 6485 df-aiota 47095 df-dfat 47129 df-afv 47130 |
| This theorem is referenced by: afveq1 47144 afveq2 47145 csbafv12g 47147 afvco2 47186 aoveq123d 47188 |
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