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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 2629 | . 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)) | |
2 | raleqd.1 | . . 3 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) | |
3 | 2 | ralbidv 2438 | . 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓)) |
4 | 1, 3 | bitrd 187 | 1 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 ∀wral 2417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 |
This theorem is referenced by: frforeq2 4275 weeq2 4287 peano5 4520 isoeq4 5713 exmidomni 7022 pitonn 7680 peano1nnnn 7684 peano2nnnn 7685 peano5nnnn 7724 peano5nni 8747 1nn 8755 peano2nn 8756 dfuzi 9185 istopg 12205 isbasisg 12250 basis2 12254 eltg2 12261 ispsmet 12531 ismet 12552 isxmet 12553 metrest 12714 cncfval 12767 bj-indeq 13298 bj-nntrans 13320 |
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