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Mirrors > Home > MPE Home > Th. List > predeq2 | Structured version Visualization version GIF version |
Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.) |
Ref | Expression |
---|---|
predeq2 | ⊢ (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . 2 ⊢ 𝑅 = 𝑅 | |
2 | eqid 2798 | . 2 ⊢ 𝑋 = 𝑋 | |
3 | predeq123 6117 | . 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑋) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋)) | |
4 | 1, 2, 3 | mp3an13 1449 | 1 ⊢ (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 Predcpred 6115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 |
This theorem is referenced by: wrecseq123 7931 wfrlem5 7942 prednn 13025 prednn0 13026 trpredeq2 33173 frmin 33197 fprlem1 33250 frrlem15 33255 |
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