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Mirrors > Home > NFE Home > Th. List > sbceq2g | GIF version |
Description: Move proper substitution to second argument of an equality. (Contributed by NM, 30-Nov-2005.) |
Ref | Expression |
---|---|
sbceq2g | ⊢ (A ∈ V → ([̣A / x]̣B = C ↔ B = [A / x]C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbceqg 3152 | . 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 | eqeq1d 2361 | . 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 = wceq 1642 ∈ wcel 1710 [̣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-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: csbsng 3785 eqerlem 5960 |
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