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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 6673 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝐺‘𝑋) = (𝐺‘𝑌)) | |
3 | 2 | fvoveq1d 7181 | . . 3 ⊢ (𝑋 = 𝑌 → (𝐹‘((𝐺‘𝑋) · 𝑂)) = (𝐹‘((𝐺‘𝑌) · 𝑂))) |
4 | 3 | breq1d 5079 | . 2 ⊢ (𝑋 = 𝑌 → ((𝐹‘((𝐺‘𝑋) · 𝑂))𝑅𝐴 ↔ (𝐹‘((𝐺‘𝑌) · 𝑂))𝑅𝐴)) |
5 | 1, 4 | imbi12d 347 | 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 208 = wceq 1536 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rab 3150 df-v 3499 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-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 |
This theorem is referenced by: rlim2 14856 rlimclim1 14905 rlimcn1 14948 climcn1 14951 caucvgrlem 15032 cncfco 23518 ftc1lem4 24639 ftc1lem6 24641 itg2gt0cn 34951 ftc1cnnclem 34969 ftc1cnnc 34970 idlimc 41913 limcperiod 41915 limclner 41938 cncfshift 42163 cncfperiod 42168 fperdvper 42209 |
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