Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cbvabwOLD | Structured version Visualization version GIF version |
Description: Obsolete version of cbvabw 2805 as of 23-May-2024. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvabwOLD.1 | ⊢ Ⅎ𝑦𝜑 |
cbvabwOLD.2 | ⊢ Ⅎ𝑥𝜓 |
cbvabwOLD.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvabwOLD | ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvabwOLD.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | sbco2v 2333 | . . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) |
3 | cbvabwOLD.2 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
4 | cbvabwOLD.3 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | sbiev 2315 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
6 | 5 | sbbii 2084 | . . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
7 | 2, 6 | bitr3i 280 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
8 | df-clab 2715 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑) | |
9 | df-clab 2715 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
10 | 7, 8, 9 | 3bitr4i 306 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ 𝑧 ∈ {𝑦 ∣ 𝜓}) |
11 | 10 | eqriv 2733 | 1 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 Ⅎwnf 1791 [wsb 2072 ∈ wcel 2112 {cab 2714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |