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Mathbox for Alexander van der Vekens |
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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 6892. (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 46403 | . . 3 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) |
4 | 1, 2 | fveq12d 6892 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐵)) |
5 | 3, 4 | ifbieq1d 4547 | . 2 ⊢ (𝜑 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = if(𝐺 defAt 𝐵, (𝐺‘𝐵), V)) |
6 | dfafv2 46409 | . 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
7 | dfafv2 46409 | . 2 ⊢ (𝐺'''𝐵) = if(𝐺 defAt 𝐵, (𝐺‘𝐵), V) | |
8 | 5, 6, 7 | 3eqtr4g 2791 | 1 ⊢ (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Vcvv 3468 ifcif 4523 ‘cfv 6537 defAt wdfat 46393 '''cafv 46394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-res 5681 df-iota 6489 df-fun 6539 df-fv 6545 df-aiota 46362 df-dfat 46396 df-afv 46397 |
This theorem is referenced by: afveq1 46411 afveq2 46412 csbafv12g 46414 afvco2 46453 aoveq123d 46455 |
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