![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ralbidv2 | Structured version Visualization version GIF version |
Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Apr-1997.) |
Ref | Expression |
---|---|
ralbidv2.1 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒))) |
Ref | Expression |
---|---|
ralbidv2 | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralbidv2.1 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒))) | |
2 | 1 | albidv 1923 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒))) |
3 | df-ral 3062 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
4 | df-ral 3062 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒)) | |
5 | 2, 3, 4 | 3bitr4g 313 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1539 ∈ wcel 2106 ∀wral 3061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 |
This theorem depends on definitions: df-bi 206 df-ral 3062 |
This theorem is referenced by: ralbidva 3175 raleqbidv 3342 ralss 4054 oneqmini 6416 ordunisuc2 7835 dfsmo2 8349 wemapsolem 9547 zorn2lem1 10493 raluz 12882 limsupgle 15423 ello12 15462 elo12 15473 lo1resb 15510 rlimresb 15511 o1resb 15512 isprm3 16622 isprm7 16647 ist1-2 22858 hausdiag 23156 xkopt 23166 cnflf 23513 cnfcf 23553 metcnp 24057 caucfil 24807 mdegleb 25589 islinds5 32525 islbs5 32541 eulerpartlemgvv 33444 filnetlem4 35352 mnuunid 43118 hoidmvle 45395 elbigo2 47316 ralbidb 47563 ralbidc 47564 |
Copyright terms: Public domain | W3C validator |