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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvifeq | Structured version Visualization version GIF version |
Description: Equality of function values with conditional arguments, see also fvif 6856. (Contributed by Alexander van der Vekens, 21-May-2018.) |
Ref | Expression |
---|---|
fvifeq | ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹‘𝐴) = if(𝜑, (𝐹‘𝐵), (𝐹‘𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6840 | . 2 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹‘𝐴) = (𝐹‘if(𝜑, 𝐵, 𝐶))) | |
2 | fvif 6856 | . 2 ⊢ (𝐹‘if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐹‘𝐵), (𝐹‘𝐶)) | |
3 | 1, 2 | eqtrdi 2792 | 1 ⊢ (𝐴 = if(𝜑, 𝐵, 𝐶) → (𝐹‘𝐴) = if(𝜑, (𝐹‘𝐵), (𝐹‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ifcif 4485 ‘cfv 6494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-iota 6446 df-fv 6502 |
This theorem is referenced by: (None) |
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