Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cbvralv | GIF version |
Description: Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.) |
Ref | Expression |
---|---|
cbvralv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cbvralv | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1421 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfv 1421 | . 2 ⊢ Ⅎ𝑥𝜓 | |
3 | cbvralv.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 1, 2, 3 | cbvral 2529 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wral 2306 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 |
This theorem is referenced by: cbvral2v 2541 cbvral3v 2543 reu7 2736 reusv3i 4191 cnvpom 4860 f1mpt 5410 grprinvlem 5695 grprinvd 5696 tfrlem1 5923 tfrlemiubacc 5944 tfrlemi1 5946 rdgon 5973 nneneq 6320 cauappcvgprlemladdrl 6755 caucvgprlemcl 6774 caucvgprlemladdrl 6776 caucvgsrlembound 6878 caucvgsrlemgt1 6879 caucvgsrlemoffres 6884 peano5nnnn 6966 axcaucvglemres 6973 nnsub 7952 ublbneg 8548 iseqovex 9219 iseqval 9220 monoord2 9236 caucvgre 9580 cvg1nlemcau 9583 resqrexlemglsq 9620 resqrexlemsqa 9622 resqrexlemex 9623 cau3lem 9710 sqrt2irr 9878 |
Copyright terms: Public domain | W3C validator |