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Mirrors > Home > MPE Home > Th. List > cbvralsv | Structured version Visualization version GIF version |
Description: Change bound variable by using a substitution. Usage of this theorem is discouraged because it depends on ax-13 2367. Use the weaker cbvralsvw 3310 when possible. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvralsv | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1910 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | nfs1v 2146 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
3 | sbequ12 2239 | . . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvral 3354 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑) |
5 | nfv 1910 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
6 | 5 | nfsb 2518 | . . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑 |
7 | nfv 1910 | . . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 | |
8 | sbequ 2079 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
9 | 6, 7, 8 | cbvral 3354 | . 2 ⊢ (∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
10 | 4, 9 | bitri 275 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 [wsb 2060 ∀wral 3057 |
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-10 2130 ax-11 2147 ax-12 2167 ax-13 2367 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clel 2806 df-nfc 2881 df-ral 3058 |
This theorem is referenced by: (None) |
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