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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 2390. Use the weaker cbvralsvw 3469 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 1915 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | nfs1v 2160 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
3 | sbequ12 2253 | . . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvral 3447 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑) |
5 | nfv 1915 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
6 | 5 | nfsb 2565 | . . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑 |
7 | nfv 1915 | . . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 | |
8 | sbequ 2090 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
9 | 6, 7, 8 | cbvral 3447 | . 2 ⊢ (∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
10 | 4, 9 | bitri 277 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 [wsb 2069 ∀wral 3140 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-10 2145 ax-11 2161 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clel 2895 df-nfc 2965 df-ral 3145 |
This theorem is referenced by: (None) |
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