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Mirrors > Home > MPE Home > Th. List > predeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.) |
Ref | Expression |
---|---|
predeq1 | ⊢ (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . 2 ⊢ 𝐴 = 𝐴 | |
2 | eqid 2738 | . 2 ⊢ 𝑋 = 𝑋 | |
3 | predeq123 6177 | . 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑋) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋)) | |
4 | 1, 2, 3 | mp3an23 1455 | 1 ⊢ (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑆, 𝐴, 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 Predcpred 6175 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3071 df-v 3423 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-op 4563 df-br 5069 df-opab 5131 df-xp 5572 df-cnv 5574 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-pred 6176 |
This theorem is referenced by: wrecseq123 8069 trpredeq1 9350 |
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