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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. (Contributed by Jeff Hankins, 19-Sep-2009.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) |
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 2926 | . . 3 ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓} |
5 | 4 | eleq2i 2872 | . 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) |
6 | df-sbc 3702 | . 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) | |
7 | df-sbc 3702 | . 2 ⊢ ([𝐴 / 𝑦]𝜓 ↔ 𝐴 ∈ {𝑦 ∣ 𝜓}) | |
8 | 5, 6, 7 | 3bitr4i 304 | 1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ [𝐴 / 𝑦]𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 Ⅎwnf 1763 ∈ wcel 2079 {cab 2773 [wsbc 3701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-clab 2774 df-cleq 2786 df-clel 2861 df-sbc 3702 |
This theorem is referenced by: cbvsbcv 3731 cbvcsb 3816 |
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