Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > mpteq2dv | GIF version |
Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 23-Aug-2014.) |
Ref | Expression |
---|---|
mpteq2dv.1 | ⊢ (𝜑 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
mpteq2dv | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpteq2dv.1 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
2 | 1 | adantr 274 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
3 | 2 | mpteq2dva 4018 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ↦ cmpt 3989 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-ral 2421 df-opab 3990 df-mpt 3991 |
This theorem is referenced by: ofeq 5984 rdgeq1 6268 rdgeq2 6269 omv 6351 oeiv 6352 0tonninf 10212 1tonninf 10213 iseqf1olemjpcl 10268 iseqf1olemqpcl 10269 iseqf1olemfvp 10270 seq3f1olemqsum 10273 seq3f1olemp 10275 summodc 11152 zsumdc 11153 fsum3 11156 prodeq2w 11325 prodmodc 11347 sloteq 11964 cnprcl2k 12375 fsumcncntop 12725 expcncf 12761 dvexp 12844 dvexp2 12845 peano4nninf 13200 peano3nninf 13201 nninfalllem1 13203 nninfsellemdc 13206 nninfsellemeq 13210 nninfsellemqall 13211 nninfsellemeqinf 13212 nninfomni 13215 |
Copyright terms: Public domain | W3C validator |