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Mirrors > Home > MPE Home > Th. List > Mathboxes > wsuceq2 | Structured version Visualization version GIF version |
Description: Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) |
Ref | Expression |
---|---|
wsuceq2 | ⊢ (𝐴 = 𝐵 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐵, 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . 2 ⊢ 𝑅 = 𝑅 | |
2 | eqid 2734 | . 2 ⊢ 𝑋 = 𝑋 | |
3 | wsuceq123 35795 | . 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑋) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐵, 𝑋)) | |
4 | 1, 2, 3 | mp3an13 1451 | 1 ⊢ (𝐴 = 𝐵 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐵, 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 wsuccwsuc 35791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-xp 5694 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-sup 9479 df-inf 9480 df-wsuc 35793 |
This theorem is referenced by: (None) |
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