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Mirrors > Home > MPE Home > Th. List > imbrov2fvoveq | Structured version Visualization version GIF version |
Description: Equality theorem for nested function and operation value in an implication for a binary relation. Technical theorem to be used to reduce the size of a significant number of proofs. (Contributed by AV, 17-Aug-2022.) |
Ref | Expression |
---|---|
imbrov2fvoveq.1 | ⊢ (𝑋 = 𝑌 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
imbrov2fvoveq | ⊢ (𝑋 = 𝑌 → ((𝜑 → (𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴) ↔ (𝜓 → (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imbrov2fvoveq.1 | . 2 ⊢ (𝑋 = 𝑌 → (𝜑 ↔ 𝜓)) | |
2 | fveq2 6907 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝐺‘𝑋) = (𝐺‘𝑌)) | |
3 | 2 | fvoveq1d 7453 | . . 3 ⊢ (𝑋 = 𝑌 → (𝐹‘((𝐺‘𝑋) · 𝑂)) = (𝐹‘((𝐺‘𝑌) · 𝑂))) |
4 | 3 | breq1d 5158 | . 2 ⊢ (𝑋 = 𝑌 → ((𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴 ↔ (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴)) |
5 | 1, 4 | imbi12d 344 | 1 ⊢ (𝑋 = 𝑌 → ((𝜑 → (𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴) ↔ (𝜓 → (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 |
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-ext 2706 |
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-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 |
This theorem is referenced by: rlim2 15529 rlimclim1 15578 rlimcn1 15621 climcn1 15625 caucvgrlem 15706 cncfco 24947 ftc1lem4 26095 ftc1lem6 26097 itg2gt0cn 37662 ftc1cnnclem 37678 ftc1cnnc 37679 idlimc 45582 limcperiod 45584 limclner 45607 cncfshift 45830 cncfperiod 45835 fperdvper 45875 |
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