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Mirrors > Home > MPE Home > Th. List > eqsbc3r | Structured version Visualization version GIF version |
Description: eqsbc3 3442 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 3442 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | |
2 | eqcom 2617 | . . 3 ⊢ (𝐵 = 𝑥 ↔ 𝑥 = 𝐵) | |
3 | 2 | sbcbii 3458 | . 2 ⊢ ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ [𝐴 / 𝑥]𝑥 = 𝐵) |
4 | eqcom 2617 | . 2 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
5 | 1, 3, 4 | 3bitr4g 302 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ 𝐵 = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 [wsbc 3402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 df-sbc 3403 |
This theorem is referenced by: sbcoreleleq 37766 sbcoreleleqVD 38117 |
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