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Mirrors > Home > MPE Home > Th. List > clelsb3vOLD | Structured version Visualization version GIF version |
Description: Obsolete version of clelsb3 2939 as of 29-Jul-2023. (Contributed by Wolf Lammen, 30-Apr-2023.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
clelsb3vOLD | ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbco2vv 2107 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑤]𝑤 ∈ 𝐴) | |
2 | eleq1w 2894 | . . . 4 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
3 | 2 | sbievw 2102 | . . 3 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴) |
4 | 3 | sbbii 2080 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴) |
5 | eleq1w 2894 | . . 3 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
6 | 5 | sbievw 2102 | . 2 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
7 | 1, 4, 6 | 3bitr3i 303 | 1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 [wsb 2068 ∈ wcel 2113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-sb 2069 df-clel 2892 |
This theorem is referenced by: (None) |
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