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Mirrors > Home > NFE Home > Th. List > csbeq1 | GIF version |
Description: Analog of dfsbcq 3048 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 3048 | . . 3 ⊢ (A = B → ([̣A / x]̣y ∈ C ↔ [̣B / x]̣y ∈ C)) | |
2 | 1 | abbidv 2467 | . 2 ⊢ (A = B → {y ∣ [̣A / x]̣y ∈ C} = {y ∣ [̣B / x]̣y ∈ C}) |
3 | df-csb 3137 | . 2 ⊢ [A / x]C = {y ∣ [̣A / x]̣y ∈ C} | |
4 | df-csb 3137 | . 2 ⊢ [B / x]C = {y ∣ [̣B / x]̣y ∈ C} | |
5 | 2, 3, 4 | 3eqtr4g 2410 | 1 ⊢ (A = B → [A / x]C = [B / x]C) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∈ wcel 1710 {cab 2339 [̣wsbc 3046 [csb 3136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-sbc 3047 df-csb 3137 |
This theorem is referenced by: csbeq1d 3142 csbeq1a 3144 csbiebg 3175 sbcnestgf 3183 cbvralcsf 3198 cbvreucsf 3200 cbvrabcsf 3201 csbing 3462 csbifg 3690 csbiotag 4371 csbopabg 4637 sbcbrg 4685 csbima12g 4955 csbovg 5552 fvmpts 5701 fvmpt2i 5703 fvmptex 5721 |
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