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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj602 | Structured version Visualization version GIF version |
Description: Equality theorem for the pred function constant. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj602 | ⊢ (𝑋 = 𝑌 → pred(𝑋, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 4845 | . . 3 ⊢ (𝑋 = 𝑌 → (𝑦𝑅𝑋 ↔ 𝑦𝑅𝑌)) | |
2 | 1 | rabbidv 3371 | . 2 ⊢ (𝑋 = 𝑌 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑌}) |
3 | df-bnj14 31266 | . 2 ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} | |
4 | df-bnj14 31266 | . 2 ⊢ pred(𝑌, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑌} | |
5 | 2, 3, 4 | 3eqtr4g 2856 | 1 ⊢ (𝑋 = 𝑌 → pred(𝑋, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 {crab 3091 class class class wbr 4841 predc-bnj14 31265 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ral 3092 df-rab 3096 df-v 3385 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-br 4842 df-bnj14 31266 |
This theorem is referenced by: bnj601 31498 bnj852 31499 bnj18eq1 31505 bnj938 31515 bnj1125 31568 bnj1148 31572 bnj1318 31601 bnj1442 31625 bnj1450 31626 bnj1452 31628 bnj1463 31631 bnj1529 31646 |
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