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Mirrors > Home > ILE Home > Th. List > csbeq1 | GIF version |
Description: Analog of dfsbcq 2948 for proper substitution into a class. (Contributed by NM, 10-Nov-2005.) |
Ref | Expression |
---|---|
csbeq1 | ⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 2948 | . . 3 ⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝑦 ∈ 𝐶 ↔ [𝐵 / 𝑥]𝑦 ∈ 𝐶)) | |
2 | 1 | abbidv 2282 | . 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶} = {𝑦 ∣ [𝐵 / 𝑥]𝑦 ∈ 𝐶}) |
3 | df-csb 3041 | . 2 ⊢ ⦋𝐴 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐴 / 𝑥]𝑦 ∈ 𝐶} | |
4 | df-csb 3041 | . 2 ⊢ ⦋𝐵 / 𝑥⦌𝐶 = {𝑦 ∣ [𝐵 / 𝑥]𝑦 ∈ 𝐶} | |
5 | 2, 3, 4 | 3eqtr4g 2222 | 1 ⊢ (𝐴 = 𝐵 → ⦋𝐴 / 𝑥⦌𝐶 = ⦋𝐵 / 𝑥⦌𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 {cab 2150 [wsbc 2946 ⦋csb 3040 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-sbc 2947 df-csb 3041 |
This theorem is referenced by: csbeq1d 3047 csbeq1a 3049 csbiebg 3082 sbcnestgf 3091 cbvralcsf 3102 cbvrexcsf 3103 cbvreucsf 3104 cbvrabcsf 3105 csbing 3324 disjnims 3968 sbcbrg 4030 csbopabg 4054 pofun 4284 csbima12g 4959 csbiotag 5175 fvmpts 5558 fvmpt2 5563 mptfvex 5565 elfvmptrab1 5574 fmptcof 5646 fmptcos 5647 fliftfuns 5760 csbriotag 5804 csbov123g 5871 eqerlem 6523 qliftfuns 6576 summodclem2a 11308 zsumdc 11311 fsum3 11314 sumsnf 11336 sumsns 11342 fsum2dlemstep 11361 fisumcom2 11365 fsumshftm 11372 fisum0diag2 11374 fsumiun 11404 prodsnf 11519 fprodm1s 11528 fprodp1s 11529 prodsns 11530 fprod2dlemstep 11549 fprodcom2fi 11553 pcmptdvds 12254 ctiunctlemf 12314 mulcncflem 13137 |
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