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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 6914. (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 47076 | . . 3 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) |
4 | 1, 2 | fveq12d 6914 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐵)) |
5 | 3, 4 | ifbieq1d 4555 | . 2 ⊢ (𝜑 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = if(𝐺 defAt 𝐵, (𝐺‘𝐵), V)) |
6 | dfafv2 47082 | . 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
7 | dfafv2 47082 | . 2 ⊢ (𝐺'''𝐵) = if(𝐺 defAt 𝐵, (𝐺‘𝐵), V) | |
8 | 5, 6, 7 | 3eqtr4g 2800 | 1 ⊢ (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Vcvv 3478 ifcif 4531 ‘cfv 6563 defAt wdfat 47066 '''cafv 47067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-res 5701 df-iota 6516 df-fun 6565 df-fv 6571 df-aiota 47035 df-dfat 47069 df-afv 47070 |
This theorem is referenced by: afveq1 47084 afveq2 47085 csbafv12g 47087 afvco2 47126 aoveq123d 47128 |
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