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Mirrors > Home > ILE Home > Th. List > infeq123d | GIF version |
Description: Equality deduction for infimum. (Contributed by AV, 2-Sep-2020.) |
Ref | Expression |
---|---|
infeq123d.a | ⊢ (𝜑 → 𝐴 = 𝐷) |
infeq123d.b | ⊢ (𝜑 → 𝐵 = 𝐸) |
infeq123d.c | ⊢ (𝜑 → 𝐶 = 𝐹) |
Ref | Expression |
---|---|
infeq123d | ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infeq123d.a | . . 3 ⊢ (𝜑 → 𝐴 = 𝐷) | |
2 | infeq123d.b | . . 3 ⊢ (𝜑 → 𝐵 = 𝐸) | |
3 | infeq123d.c | . . . 4 ⊢ (𝜑 → 𝐶 = 𝐹) | |
4 | 3 | cnveqd 4529 | . . 3 ⊢ (𝜑 → ◡𝐶 = ◡𝐹) |
5 | 1, 2, 4 | supeq123d 6404 | . 2 ⊢ (𝜑 → sup(𝐴, 𝐵, ◡𝐶) = sup(𝐷, 𝐸, ◡𝐹)) |
6 | df-inf 6398 | . 2 ⊢ inf(𝐴, 𝐵, 𝐶) = sup(𝐴, 𝐵, ◡𝐶) | |
7 | df-inf 6398 | . 2 ⊢ inf(𝐷, 𝐸, 𝐹) = sup(𝐷, 𝐸, ◡𝐹) | |
8 | 5, 6, 7 | 3eqtr4g 2138 | 1 ⊢ (𝜑 → inf(𝐴, 𝐵, 𝐶) = inf(𝐷, 𝐸, 𝐹)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ◡ccnv 4362 supcsup 6395 infcinf 6396 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-in 2979 df-ss 2986 df-uni 3602 df-br 3786 df-opab 3840 df-cnv 4371 df-sup 6397 df-inf 6398 |
This theorem is referenced by: (None) |
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