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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 2379. Use the weaker cbvralw 3387 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 2955 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2955 | . 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 3385 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 Ⅎwnf 1785 ∀wral 3106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-10 2142 ax-11 2158 ax-12 2175 ax-13 2379 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clel 2870 df-nfc 2938 df-ral 3111 |
This theorem is referenced by: cbvralv 3399 cbvralsv 3416 cbviing 4926 ralrnmpt 6839 |
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