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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 2739 | . 2 ⊢ 𝑅 = 𝑅 | |
2 | eqid 2739 | . 2 ⊢ 𝑋 = 𝑋 | |
3 | predeq123 6131 | . 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑋) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋)) | |
4 | 1, 2, 3 | mp3an13 1453 | 1 ⊢ (𝐴 = 𝐵 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐵, 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 Predcpred 6129 |
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 2711 |
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 2718 df-cleq 2731 df-clel 2812 df-rab 3063 df-v 3401 df-un 3849 df-in 3851 df-ss 3861 df-sn 4518 df-pr 4520 df-op 4524 df-br 5032 df-opab 5094 df-xp 5532 df-cnv 5534 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 |
This theorem is referenced by: wrecseq123 7980 wfrlem5 7991 prednn 13124 prednn0 13125 trpredeq2 33368 frmin 33395 fprlem1 33460 frrlem15 33465 |
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