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Mirrors > Home > MPE Home > Th. List > cbvralsvw | Structured version Visualization version GIF version |
Description: Change bound variable by using a substitution. Version of cbvralsv 3466 with a disjoint variable condition, which does not require ax-13 2389. (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 1914 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
2 | nfs1v 2159 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑 | |
3 | sbequ12 2252 | . . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑)) | |
4 | 1, 2, 3 | cbvralw 3438 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑) |
5 | nfv 1914 | . . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑 | |
6 | nfv 1914 | . . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑 | |
7 | sbequ 2089 | . . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
8 | 5, 6, 7 | cbvralw 3438 | . 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 2068 ∀wral 3137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-10 2144 ax-11 2160 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-nf 1784 df-sb 2069 df-clel 2892 df-nfc 2962 df-ral 3142 |
This theorem is referenced by: sbralie 3468 rspsbc 3856 ralxpf 5710 tfinds 7567 tfindes 7570 nn0min 30534 |
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