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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 6909. (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 46535 | . . 3 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) |
4 | 1, 2 | fveq12d 6909 | . . 3 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐵)) |
5 | 3, 4 | ifbieq1d 4556 | . 2 ⊢ (𝜑 → if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) = if(𝐺 defAt 𝐵, (𝐺‘𝐵), V)) |
6 | dfafv2 46541 | . 2 ⊢ (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹‘𝐴), V) | |
7 | dfafv2 46541 | . 2 ⊢ (𝐺'''𝐵) = if(𝐺 defAt 𝐵, (𝐺‘𝐵), V) | |
8 | 5, 6, 7 | 3eqtr4g 2793 | 1 ⊢ (𝜑 → (𝐹'''𝐴) = (𝐺'''𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Vcvv 3473 ifcif 4532 ‘cfv 6553 defAt wdfat 46525 '''cafv 46526 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-res 5694 df-iota 6505 df-fun 6555 df-fv 6561 df-aiota 46494 df-dfat 46528 df-afv 46529 |
This theorem is referenced by: afveq1 46543 afveq2 46544 csbafv12g 46546 afvco2 46585 aoveq123d 46587 |
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