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Mirrors > Home > MPE Home > Th. List > wrecseq3 | Structured version Visualization version GIF version |
Description: Equality theorem for the well-ordered recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.) |
Ref | Expression |
---|---|
wrecseq3 | ⊢ (𝐹 = 𝐺 → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑅, 𝐴, 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . 2 ⊢ 𝑅 = 𝑅 | |
2 | eqid 2728 | . 2 ⊢ 𝐴 = 𝐴 | |
3 | wrecseq123 8326 | . 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑅, 𝐴, 𝐺)) | |
4 | 1, 2, 3 | mp3an12 1447 | 1 ⊢ (𝐹 = 𝐺 → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑅, 𝐴, 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 wrecscwrecs 8323 |
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-ext 2699 |
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-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rab 3431 df-v 3475 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-br 5153 df-opab 5215 df-xp 5688 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-iota 6505 df-fv 6561 df-ov 7429 df-frecs 8293 df-wrecs 8324 |
This theorem is referenced by: recseq 8401 bpolylem 16032 |
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