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Mirrors > Home > MPE Home > Th. List > cbvsbc | Structured version Visualization version GIF version |
Description: Change bound variables in a wff substitution. Usage of this theorem is discouraged because it depends on ax-13 2366. Use the weaker cbvsbcw 3810 when possible. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvsbc.1 | ⊢ Ⅎ𝑦𝜑 |
cbvsbc.2 | ⊢ Ⅎ𝑥𝜓 |
cbvsbc.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvsbc | ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbvsbc.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
2 | cbvsbc.2 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvsbc.3 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvab 2801 | . . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
5 | 4 | eleq2i 2818 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) |
6 | df-sbc 3777 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
7 | df-sbc 3777 | . 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) | |
8 | 5, 6, 7 | 3bitr4i 302 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 Ⅎwnf 1778 ∈ wcel 2099 {cab 2703 [wsbc 3776 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-13 2366 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-sbc 3777 |
This theorem is referenced by: cbvsbcv 3813 cbvcsb 3903 |
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