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Mirrors > Home > ILE Home > Th. List > raleqbi1dv | GIF version |
Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.) |
Ref | Expression |
---|---|
raleqd.1 | ⊢ (A = B → (φ ↔ ψ)) |
Ref | Expression |
---|---|
raleqbi1dv | ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | raleq 2499 | . 2 ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B φ)) | |
2 | raleqd.1 | . . 3 ⊢ (A = B → (φ ↔ ψ)) | |
3 | 2 | ralbidv 2320 | . 2 ⊢ (A = B → (∀x ∈ B φ ↔ ∀x ∈ B ψ)) |
4 | 1, 3 | bitrd 177 | 1 ⊢ (A = B → (∀x ∈ A φ ↔ ∀x ∈ B ψ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 = wceq 1242 ∀wral 2300 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 |
This theorem is referenced by: peano5 4264 isoeq4 5387 pitonn 6744 peano5nni 7698 1nn 7706 peano2nn 7707 dfuzi 8124 bj-indeq 9388 bj-nntrans 9411 |
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