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Mirrors > Home > MPE Home > Th. List > predeq3 | Structured version Visualization version GIF version |
Description: Equality theorem for the predecessor class. (Contributed by Scott Fenton, 2-Feb-2011.) |
Ref | Expression |
---|---|
predeq3 | ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . 2 ⊢ 𝑅 = 𝑅 | |
2 | eqid 2622 | . 2 ⊢ 𝐴 = 𝐴 | |
3 | predeq123 5681 | . 2 ⊢ ((𝑅 = 𝑅 ∧ 𝐴 = 𝐴 ∧ 𝑋 = 𝑌) → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌)) | |
4 | 1, 2, 3 | mp3an12 1414 | 1 ⊢ (𝑋 = 𝑌 → Pred(𝑅, 𝐴, 𝑋) = Pred(𝑅, 𝐴, 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 Predcpred 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 |
This theorem is referenced by: dfpred3g 5691 predbrg 5700 preddowncl 5707 wfisg 5715 wfr3g 7413 wfrlem1 7414 wfrdmcl 7423 wfrlem14 7428 wfrlem15 7429 wfrlem17 7431 wfr2a 7432 trpredeq3 31722 trpredlem1 31727 trpredtr 31730 trpredmintr 31731 trpredrec 31738 frmin 31739 frinsg 31742 elwlim 31769 elwlimOLD 31770 frr3g 31779 frrlem1 31780 frrlem5e 31788 csbwrecsg 33173 |
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