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Mirrors > Home > ILE Home > Th. List > rabbidv | GIF version |
Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (Contributed by NM, 10-Feb-1995.) |
Ref | Expression |
---|---|
rabbidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
rabbidv | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabbidv.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | adantr 276 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
3 | 2 | rabbidva 2725 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐴 ∣ 𝜒}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2148 {crab 2459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-ral 2460 df-rab 2464 |
This theorem is referenced by: rabeqbidv 2732 difeq2 3247 seex 4331 mptiniseg 5118 supeq1 6978 supeq2 6981 supeq3 6982 cardcl 7173 isnumi 7174 cardval3ex 7177 carden2bex 7181 genpdflem 7484 genipv 7486 genpelxp 7488 addcomprg 7555 mulcomprg 7557 uzval 9506 ixxval 9870 fzval 9984 hashinfom 10729 hashennn 10731 shftfn 10804 gcdval 11930 lcmval 12033 isprm 12079 odzval 12211 pceulem 12264 pceu 12265 pcval 12266 pczpre 12267 pcdiv 12272 istopon 13144 toponsspwpwg 13153 clsval 13244 neival 13276 cnpval 13331 blvalps 13521 blval 13522 limccl 13761 ellimc3apf 13762 eldvap 13784 |
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