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Mirrors > Home > NFE Home > Th. List > eqsbc1 | GIF version |
Description: Substitution for the left-hand side in an equality. Class version of eqsb1 2454. (Contributed by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
eqsbc1 | ⊢ (A ∈ V → ([̣A / x]̣x = B ↔ A = B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsbcq 3049 | . 2 ⊢ (y = A → ([̣y / x]̣x = B ↔ [̣A / x]̣x = B)) | |
2 | eqeq1 2359 | . 2 ⊢ (y = A → (y = B ↔ A = B)) | |
3 | sbsbc 3051 | . . 3 ⊢ ([y / x]x = B ↔ [̣y / x]̣x = B) | |
4 | eqsb1 2454 | . . 3 ⊢ ([y / x]x = B ↔ y = B) | |
5 | 3, 4 | bitr3i 242 | . 2 ⊢ ([̣y / x]̣x = B ↔ y = B) |
6 | 1, 2, 5 | vtoclbg 2916 | 1 ⊢ (A ∈ V → ([̣A / x]̣x = B ↔ A = B)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 = wceq 1642 [wsb 1648 ∈ wcel 1710 [̣wsbc 3047 |
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 2479 df-v 2862 df-sbc 3048 |
This theorem is referenced by: sbceqal 3098 eqsbc2 3104 |
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