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| Mirrors > Home > MPE Home > Th. List > raleqbii | Structured version Visualization version GIF version | ||
| Description: Equality deduction for restricted universal quantifier, changing both formula and quantifier domain. Inference form. (Contributed by David Moews, 1-May-2017.) | 
| Ref | Expression | 
|---|---|
| raleqbii.1 | ⊢ 𝐴 = 𝐵 | 
| raleqbii.2 | ⊢ (𝜓 ↔ 𝜒) | 
| Ref | Expression | 
|---|---|
| raleqbii | ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | raleqbii.1 | . . . 4 ⊢ 𝐴 = 𝐵 | |
| 2 | 1 | eleq2i 2833 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) | 
| 3 | raleqbii.2 | . . 3 ⊢ (𝜓 ↔ 𝜒) | |
| 4 | 2, 3 | imbi12i 350 | . 2 ⊢ ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒)) | 
| 5 | 4 | ralbii2 3089 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∀wral 3061 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-clel 2816 df-ral 3062 | 
| This theorem is referenced by: fprlem1 8325 wfrlem5OLD 8353 frrlem15 9797 opprdomnb 20717 ply1coe 22302 ordtbaslem 23196 iscusp2 24311 isrgr 29577 iineq12i 36198 elghomOLD 37894 iscrngo2 38004 tendoset 40761 comptiunov2i 43719 | 
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