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 6680. (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 43332 | . . 3 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) |
4 | 1, 2 | fveq12d 6680 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐵)) |
5 | 3, 4 | ifbieq1d 4493 | . 2 ⊢ (𝜑 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = if(𝐺 defAt 𝐵, (𝐺‘𝐵), V)) |
6 | dfafv2 43338 | . 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
7 | dfafv2 43338 | . 2 ⊢ (𝐺'''𝐵) = if(𝐺 defAt 𝐵, (𝐺‘𝐵), V) | |
8 | 5, 6, 7 | 3eqtr4g 2884 | 1 ⊢ (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 Vcvv 3497 ifcif 4470 ‘cfv 6358 defAt wdfat 43322 '''cafv 43323 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-int 4880 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-res 5570 df-iota 6317 df-fun 6360 df-fv 6366 df-aiota 43292 df-dfat 43325 df-afv 43326 |
This theorem is referenced by: afveq1 43340 afveq2 43341 csbafv12g 43343 afvco2 43382 aoveq123d 43384 |
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