![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cbvralsv | Structured version Visualization version GIF version |
Description: Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.) |
Ref | Expression |
---|---|
cbvralsv | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 2015 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | nfs1v 2313 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
3 | sbequ12 2288 | . . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvral 3379 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑) |
5 | nfv 2015 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
6 | 5 | nfsb 2575 | . . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑 |
7 | nfv 2015 | . . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 | |
8 | sbequ 2507 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
9 | 6, 7, 8 | cbvral 3379 | . 2 ⊢ (∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
10 | 4, 9 | bitri 267 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 [wsb 2069 ∀wral 3117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clel 2821 df-nfc 2958 df-ral 3122 |
This theorem is referenced by: sbralie 3396 rspsbc 3742 ralxpf 5501 tfinds 7320 tfindes 7323 nn0min 30114 |
Copyright terms: Public domain | W3C validator |