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Mirrors > Home > MPE Home > Th. List > Mathboxes > wsuceq3 | Structured version Visualization version GIF version |
Description: Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) |
Ref | Expression |
---|---|
wsuceq3 | ⊢ (𝑋 = 𝑌 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐴, 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . 2 ⊢ 𝑅 = 𝑅 | |
2 | eqid 2818 | . 2 ⊢ 𝐴 = 𝐴 | |
3 | wsuceq123 32998 | . 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐴, 𝑌)) | |
4 | 1, 2, 3 | mp3an12 1442 | 1 ⊢ (𝑋 = 𝑌 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐴, 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 wsuccwsuc 32994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-sup 8894 df-inf 8895 df-wsuc 32996 |
This theorem is referenced by: (None) |
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