![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > clelsb2OLD | Structured version Visualization version GIF version |
Description: Obsolete version of clelsb2 2860 as of 24-Nov-2024.) (Contributed by Jim Kingdon, 22-Nov-2018.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
clelsb2OLD | ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1916 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑤 | |
2 | 1 | sbco2 2509 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝐴 ∈ 𝑤 ↔ [𝑦 / 𝑤]𝐴 ∈ 𝑤) |
3 | nfv 1916 | . . . 4 ⊢ Ⅎ𝑤 𝐴 ∈ 𝑥 | |
4 | eleq2 2821 | . . . 4 ⊢ (𝑤 = 𝑥 → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑥)) | |
5 | 3, 4 | sbie 2500 | . . 3 ⊢ ([𝑥 / 𝑤]𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑥) |
6 | 5 | sbbii 2078 | . 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝐴 ∈ 𝑤 ↔ [𝑦 / 𝑥]𝐴 ∈ 𝑥) |
7 | nfv 1916 | . . 3 ⊢ Ⅎ𝑤 𝐴 ∈ 𝑦 | |
8 | eleq2 2821 | . . 3 ⊢ (𝑤 = 𝑦 → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑦)) | |
9 | 7, 8 | sbie 2500 | . 2 ⊢ ([𝑦 / 𝑤]𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑦) |
10 | 2, 6, 9 | 3bitr3i 301 | 1 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 [wsb 2066 ∈ wcel 2105 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-13 2370 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-cleq 2723 df-clel 2809 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |