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| Mirrors > Home > MPE Home > Th. List > cbvral | Structured version Visualization version GIF version | ||
| Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2377. Use the weaker cbvralw 3306 when possible. (Contributed by NM, 31-Jul-2003.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| cbvral.1 | ⊢ Ⅎ𝑦𝜑 | 
| cbvral.2 | ⊢ Ⅎ𝑥𝜓 | 
| cbvral.3 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | 
| Ref | Expression | 
|---|---|
| cbvral | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfcv 2905 | . 2 ⊢ Ⅎ𝑥𝐴 | |
| 2 | nfcv 2905 | . 2 ⊢ Ⅎ𝑦𝐴 | |
| 3 | cbvral.1 | . 2 ⊢ Ⅎ𝑦𝜑 | |
| 4 | cbvral.2 | . 2 ⊢ Ⅎ𝑥𝜓 | |
| 5 | cbvral.3 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
| 6 | 1, 2, 3, 4, 5 | cbvralf 3360 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 Ⅎwnf 1783 ∀wral 3061 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clel 2816 df-nfc 2892 df-ral 3062 | 
| This theorem is referenced by: cbvralv 3364 cbvralsv 3366 cbviing 5039 ralrnmpt 7116 | 
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