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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 6913. (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 47138 | . . 3 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) | 
| 4 | 1, 2 | fveq12d 6913 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐵)) | 
| 5 | 3, 4 | ifbieq1d 4550 | . 2 ⊢ (𝜑 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = if(𝐺 defAt 𝐵, (𝐺‘𝐵), V)) | 
| 6 | dfafv2 47144 | . 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
| 7 | dfafv2 47144 | . 2 ⊢ (𝐺'''𝐵) = if(𝐺 defAt 𝐵, (𝐺‘𝐵), V) | |
| 8 | 5, 6, 7 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 Vcvv 3480 ifcif 4525 ‘cfv 6561 defAt wdfat 47128 '''cafv 47129 | 
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-aiota 47097 df-dfat 47131 df-afv 47132 | 
| This theorem is referenced by: afveq1 47146 afveq2 47147 csbafv12g 47149 afvco2 47188 aoveq123d 47190 | 
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