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Mirrors > Home > MPE Home > Th. List > ralbida | Structured version Visualization version GIF version |
Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003.) |
Ref | Expression |
---|---|
ralbida.1 | ⊢ Ⅎ𝑥𝜑 |
ralbida.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
ralbida | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralbida.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | ralbida.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
3 | 2 | pm5.74da 802 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐴 → 𝜒))) |
4 | 1, 3 | albid 2223 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒))) |
5 | df-ral 3146 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
6 | df-ral 3146 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜒)) | |
7 | 4, 5, 6 | 3bitr4g 316 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 Ⅎwnf 1783 ∈ wcel 2113 ∀wral 3141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-12 2176 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-nf 1784 df-ral 3146 |
This theorem is referenced by: ralbid 3234 2ralbida 3235 ralbiOLD 3236 ac6num 9904 neiptopreu 21744 istrkg2ld 26249 funcnv5mpt 30416 xrralrecnnge 41668 climf2 41953 clim2f2 41957 limsupub 41991 climinfmpt 42002 limsupubuzmpt 42006 limsupre2mpt 42017 limsupre3mpt 42021 limsupreuzmpt 42026 xlimmnfmpt 42130 xlimpnfmpt 42131 smfsupmpt 43096 smfinfmpt 43100 |
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