![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cbvralv2 | Structured version Visualization version GIF version |
Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Usage of this theorem is discouraged because it depends on ax-13 2371. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cbvralv2.1 | ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) |
cbvralv2.2 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
cbvralv2 | ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2908 | . 2 ⊢ Ⅎ𝑦𝐴 | |
2 | nfcv 2908 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | nfv 1918 | . 2 ⊢ Ⅎ𝑦𝜓 | |
4 | nfv 1918 | . 2 ⊢ Ⅎ𝑥𝜒 | |
5 | cbvralv2.2 | . 2 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
6 | cbvralv2.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)) | |
7 | 1, 2, 3, 4, 5, 6 | cbvralcsf 3905 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∀wral 3065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-13 2371 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-sbc 3745 df-csb 3861 |
This theorem is referenced by: pgindnf 47235 |
Copyright terms: Public domain | W3C validator |