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Mathbox for Jonathan Ben-Naim |
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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 5107 | . . 3 ⊢ (𝑋 = 𝑌 → (𝑦𝑅𝑋 ↔ 𝑦𝑅𝑌)) | |
2 | 1 | rabbidv 3413 | . 2 ⊢ (𝑋 = 𝑌 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑌}) |
3 | df-bnj14 33113 | . 2 ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} | |
4 | df-bnj14 33113 | . 2 ⊢ pred(𝑌, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑌} | |
5 | 2, 3, 4 | 3eqtr4g 2802 | 1 ⊢ (𝑋 = 𝑌 → pred(𝑋, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 {crab 3405 class class class wbr 5103 predc-bnj14 33112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-bnj14 33113 |
This theorem is referenced by: bnj601 33344 bnj852 33345 bnj18eq1 33351 bnj938 33361 bnj1125 33416 bnj1148 33420 bnj1318 33449 bnj1442 33473 bnj1450 33474 bnj1452 33476 bnj1463 33479 bnj1529 33494 |
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