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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 2740 | . 2 ⊢ 𝑅 = 𝑅 | |
2 | eqid 2740 | . 2 ⊢ 𝑋 = 𝑋 | |
3 | wsuceq123 33796 | . 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 1542 wsuccwsuc 33792 |
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 2015 ax-8 2112 ax-9 2120 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-xp 5595 df-cnv 5597 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-sup 9171 df-inf 9172 df-wsuc 33794 |
This theorem is referenced by: (None) |
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