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Mirrors > Home > MPE Home > Th. List > wrecseq1 | 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 |
---|---|
wrecseq1 | ⊢ (𝑅 = 𝑆 → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐴, 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . 2 ⊢ 𝐴 = 𝐴 | |
2 | eqid 2733 | . 2 ⊢ 𝐹 = 𝐹 | |
3 | wrecseq123 8246 | . 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐴 ∧ 𝐹 = 𝐹) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐴, 𝐹)) | |
4 | 1, 2, 3 | mp3an23 1454 | 1 ⊢ (𝑅 = 𝑆 → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐴, 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 wrecscwrecs 8243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-xp 5640 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-iota 6449 df-fv 6505 df-ov 7361 df-frecs 8213 df-wrecs 8244 |
This theorem is referenced by: csbrecsg 35845 |
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