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Mirrors > Home > MPE Home > Th. List > wrecseq2 | Structured version Visualization version GIF version |
Description: Equality theorem for the well-founded recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.) |
Ref | Expression |
---|---|
wrecseq2 | ⊢ (𝐴 = 𝐵 → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑅, 𝐵, 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . 2 ⊢ 𝑅 = 𝑅 | |
2 | eqid 2821 | . 2 ⊢ 𝐹 = 𝐹 | |
3 | wrecseq123 7947 | . 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐹) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑅, 𝐵, 𝐹)) | |
4 | 1, 2, 3 | mp3an13 1448 | 1 ⊢ (𝐴 = 𝐵 → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑅, 𝐵, 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 wrecscwrecs 7945 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-xp 5560 df-cnv 5562 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-iota 6313 df-fv 6362 df-wrecs 7946 |
This theorem is referenced by: csbrecsg 34608 |
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