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Mirrors > Home > MPE Home > Th. List > cbvab | Structured version Visualization version GIF version |
Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2370. Usage of the weaker cbvabw 2810 and cbvabv 2809 are preferred. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvab.1 | ⊢ Ⅎ𝑦𝜑 |
cbvab.2 | ⊢ Ⅎ𝑥𝜓 |
cbvab.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvab | ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvab.1 | . . . . 5 ⊢ Ⅎ𝑦𝜑 | |
2 | 1 | sbco2 2513 | . . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥]𝜑) |
3 | cbvab.2 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
4 | cbvab.3 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
5 | 3, 4 | sbie 2504 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
6 | 5 | sbbii 2078 | . . . 4 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
7 | 2, 6 | bitr3i 276 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ [𝑧 / 𝑦]𝜓) |
8 | df-clab 2714 | . . 3 ⊢ (𝑧 ∈ {𝑥 ∣ 𝜑} ↔ [𝑧 / 𝑥]𝜑) | |
9 | df-clab 2714 | . . 3 ⊢ (𝑧 ∈ {𝑦 ∣ 𝜓} ↔ [𝑧 / 𝑦]𝜓) | |
10 | 7, 8, 9 | 3bitr4i 302 | . 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 205 = wceq 1540 Ⅎwnf 1784 [wsb 2066 ∈ wcel 2105 {cab 2713 |
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-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-13 2370 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-tru 1543 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2714 df-cleq 2728 |
This theorem is referenced by: cbvrab 3440 cdeqab1 3718 cbvsbc 3763 cbvrabcsf 3891 |
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