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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 2701 | . 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜑)) | |
| 2 | raleqd.1 | . . 3 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) | |
| 3 | 2 | ralbidv 2505 | . 2 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓)) |
| 4 | 1, 3 | bitrd 188 | 1 ⊢ (𝐴 = 𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓)) |
| Colors of variables: wff set class |
| Syntax hints: → wi 4 ↔ wb 105 = wceq 1372 ∀wral 2483 |
| 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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-ext 2186 |
| This theorem depends on definitions: df-bi 117 df-tru 1375 df-nf 1483 df-sb 1785 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 |
| This theorem is referenced by: frforeq2 4391 weeq2 4403 peano5 4645 isoeq4 5872 exmidomni 7243 tapeq2 7364 pitonn 7960 peano1nnnn 7964 peano2nnnn 7965 peano5nnnn 8004 peano5nni 9038 1nn 9046 peano2nn 9047 dfuzi 9482 mhmpropd 13269 issubm 13275 isghm 13550 ghmeql 13574 iscmn 13600 dfrhm2 13887 islssm 14090 islssmg 14091 istopg 14442 isbasisg 14487 basis2 14491 eltg2 14496 ispsmet 14766 ismet 14787 isxmet 14788 metrest 14949 cncfval 15015 bj-indeq 15827 bj-nntrans 15849 |
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