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Mirrors > Home > MPE Home > Th. List > infeq1i | Structured version Visualization version GIF version |
Description: Equality inference for infimum. (Contributed by AV, 2-Sep-2020.) |
Ref | Expression |
---|---|
infeq1i.1 | ⊢ 𝐵 = 𝐶 |
Ref | Expression |
---|---|
infeq1i | ⊢ inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infeq1i.1 | . 2 ⊢ 𝐵 = 𝐶 | |
2 | infeq1 8265 | . 2 ⊢ (𝐵 = 𝐶 → inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ inf(𝐵, 𝐴, 𝑅) = inf(𝐶, 𝐴, 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 infcinf 8230 |
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-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-rex 2902 df-rab 2905 df-uni 4373 df-sup 8231 df-inf 8232 |
This theorem is referenced by: infsn 8293 nninf 11645 nn0inf 11646 lcmcom 15144 lcmass 15165 lcmf0 15185 imasdsval2 15999 imasdsf1olem 21988 ftalem6 24604 ioodvbdlimc1lem2 38822 ioodvbdlimc2lem 38824 elaa2 39127 etransc 39176 ioorrnopn 39201 ovnval2 39435 ovolval3 39537 vonioolem2 39572 |
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