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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 2738 | . 2 ⊢ 𝑅 = 𝑅 | |
2 | eqid 2738 | . 2 ⊢ 𝐴 = 𝐴 | |
3 | wsuceq123 33421 | . 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐴, 𝑌)) | |
4 | 1, 2, 3 | mp3an12 1452 | 1 ⊢ (𝑋 = 𝑌 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐴, 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 wsuccwsuc 33417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-sb 2075 df-clab 2717 df-cleq 2730 df-clel 2811 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-un 3848 df-in 3850 df-ss 3860 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-xp 5531 df-cnv 5533 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-sup 8979 df-inf 8980 df-wsuc 33419 |
This theorem is referenced by: (None) |
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