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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 2016 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒))) |
3 | df-ral 3094 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
4 | df-ral 3094 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒)) | |
5 | 2, 3, 4 | 3bitr4g 306 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1651 ∈ wcel 2157 ∀wral 3089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 |
This theorem depends on definitions: df-bi 199 df-ral 3094 |
This theorem is referenced by: ralbidva 3166 ralss 3864 oneqmini 5992 ordunisuc2 7278 dfsmo2 7683 wemapsolem 8697 zorn2lem1 9606 raluz 11980 limsupgle 14549 ello12 14588 elo12 14599 lo1resb 14636 rlimresb 14637 o1resb 14638 isprm3 15730 isprm7 15753 ist1 21454 ist1-2 21480 hausdiag 21777 xkopt 21787 cnflf 22134 cnfcf 22174 metcnp 22674 caucfil 23409 mdegleb 24165 eulerpartlemgvv 30954 filnetlem4 32888 hoidmvle 41560 elbigo2 43145 |
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