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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 4960 | . . 3 ⊢ (𝑋 = 𝑌 → (𝑦𝑅𝑋 ↔ 𝑦𝑅𝑌)) | |
2 | 1 | rabbidv 3420 | . 2 ⊢ (𝑋 = 𝑌 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑌}) |
3 | df-bnj14 31532 | . 2 ⊢ pred(𝑋, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑋} | |
4 | df-bnj14 31532 | . 2 ⊢ pred(𝑌, 𝐴, 𝑅) = {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑌} | |
5 | 2, 3, 4 | 3eqtr4g 2854 | 1 ⊢ (𝑋 = 𝑌 → pred(𝑋, 𝐴, 𝑅) = pred(𝑌, 𝐴, 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1520 {crab 3107 class class class wbr 4956 predc-bnj14 31531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-ext 2767 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ral 3108 df-rab 3112 df-v 3434 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-sn 4467 df-pr 4469 df-op 4473 df-br 4957 df-bnj14 31532 |
This theorem is referenced by: bnj601 31764 bnj852 31765 bnj18eq1 31771 bnj938 31781 bnj1125 31834 bnj1148 31838 bnj1318 31867 bnj1442 31891 bnj1450 31892 bnj1452 31894 bnj1463 31897 bnj1529 31912 |
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