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Mirrors > Home > ILE Home > Th. List > eqsbc3r | GIF version |
Description: eqsbc3 2948 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by JJ, 7-Jul-2021.) |
Ref | Expression |
---|---|
eqsbc3r | ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ 𝐵 = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsbc3 2948 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | |
2 | eqcom 2141 | . . 3 ⊢ (𝐵 = 𝑥 ↔ 𝑥 = 𝐵) | |
3 | 2 | sbcbii 2968 | . 2 ⊢ ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ [𝐴 / 𝑥]𝑥 = 𝐵) |
4 | eqcom 2141 | . 2 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
5 | 1, 3, 4 | 3bitr4g 222 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ 𝐵 = 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 [wsbc 2909 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-sbc 2910 |
This theorem is referenced by: (None) |
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