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Mirrors > Home > MPE Home > Th. List > eqsbc3r | Structured version Visualization version GIF version |
Description: eqsbc3 3819 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 3819 | . 2 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑥 = 𝐵 ↔ 𝐴 = 𝐵)) | |
2 | eqcom 2830 | . . 3 ⊢ (𝐵 = 𝑥 ↔ 𝑥 = 𝐵) | |
3 | 2 | sbcbii 3831 | . 2 ⊢ ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ [𝐴 / 𝑥]𝑥 = 𝐵) |
4 | eqcom 2830 | . 2 ⊢ (𝐵 = 𝐴 ↔ 𝐴 = 𝐵) | |
5 | 1, 3, 4 | 3bitr4g 316 | 1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 = 𝑥 ↔ 𝐵 = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 [wsbc 3774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1540 df-ex 1781 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-sbc 3775 |
This theorem is referenced by: sbcoreleleq 40876 sbcoreleleqVD 41200 |
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