New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > sbcel2g | GIF version |
Description: Move proper substitution in and out of a membership relation. (Contributed by NM, 14-Nov-2005.) |
Ref | Expression |
---|---|
sbcel2g | ⊢ (A ∈ V → ([̣A / x]̣B ∈ C ↔ B ∈ [A / x]C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbcel12g 3151 | . 2 ⊢ (A ∈ V → ([̣A / x]̣B ∈ C ↔ [A / x]B ∈ [A / x]C)) | |
2 | csbconstg 3150 | . . 3 ⊢ (A ∈ V → [A / x]B = B) | |
3 | 2 | eleq1d 2419 | . 2 ⊢ (A ∈ V → ([A / x]B ∈ [A / x]C ↔ B ∈ [A / x]C)) |
4 | 1, 3 | bitrd 244 | 1 ⊢ (A ∈ V → ([̣A / x]̣B ∈ C ↔ B ∈ [A / x]C)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∈ wcel 1710 [̣wsbc 3046 [csb 3136 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-or 359 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-nfc 2478 df-v 2861 df-sbc 3047 df-csb 3137 |
This theorem is referenced by: csbcomg 3159 sbccsbg 3164 sbnfc2 3196 csbabg 3197 sbcss 3660 csbunig 3899 csbxpg 4813 csbrng 4966 |
Copyright terms: Public domain | W3C validator |