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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 4587 | . . 3 ⊢ (𝑋 = 𝑌 → (𝑦𝑅𝑋 ↔ 𝑦𝑅𝑌)) | |
2 | 1 | rabbidv 3164 | . 2 ⊢ (𝑋 = 𝑌 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑌}) |
3 | df-bnj14 30008 | . 2 ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} | |
4 | df-bnj14 30008 | . 2 ⊢ pred(𝑌, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑌} | |
5 | 2, 3, 4 | 3eqtr4g 2669 | 1 ⊢ (𝑋 = 𝑌 → pred(𝑋, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 {crab 2900 class class class wbr 4583 predc-bnj14 30007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-bnj14 30008 |
This theorem is referenced by: bnj601 30244 bnj852 30245 bnj18eq1 30251 bnj938 30261 bnj1125 30314 bnj1148 30318 bnj1318 30347 bnj1442 30371 bnj1450 30372 bnj1452 30374 bnj1463 30377 bnj1529 30392 |
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