Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > cbvralsvw | Structured version Visualization version GIF version |
Description: Change bound variable by using a substitution. Version of cbvralsv 3469 with a disjoint variable condition, which does not require ax-13 2390. (Contributed by NM, 20-Nov-2005.) (Revised by Gino Giotto, 10-Jan-2024.) |
Ref | Expression |
---|---|
cbvralsvw | ⊢ (∀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 | cbvralw 3441 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑) |
5 | nfv 1915 | . . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑 | |
6 | nfv 1915 | . . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 | |
7 | sbequ 2090 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
8 | 5, 6, 7 | cbvralw 3441 | . 2 ⊢ (∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
9 | 4, 8 | bitri 277 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 [wsb 2069 ∀wral 3138 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 df-clel 2893 df-nfc 2963 df-ral 3143 |
This theorem is referenced by: sbralie 3471 rspsbc 3862 ralxpf 5717 tfinds 7574 tfindes 7577 nn0min 30536 |
Copyright terms: Public domain | W3C validator |