![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > raleqi | GIF version |
Description: Equality inference for restricted universal qualifier. (Contributed by Paul Chapman, 22-Jun-2011.) |
Ref | Expression |
---|---|
raleq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
raleqi | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | raleq 2686 | . 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1364 ∀wral 2468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 |
This theorem is referenced by: ralrab2 2917 ralprg 3658 raltpg 3660 omsinds 4639 ralxp 4788 ralrnmpo 6012 nnnninfeq2 7158 fzprval 10114 fztpval 10115 seq3f1olemp 10535 zsumdc 11427 zproddc 11622 infssuzex 11985 2prm 12162 xpsfrnel 12823 nninfsellemdc 15238 nninfsellemsuc 15240 |
Copyright terms: Public domain | W3C validator |