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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 2871 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) |
3 | raleqbii.2 | . . 3 ⊢ (𝜓 ↔ 𝜒) | |
4 | 2, 3 | imbi12i 342 | . 2 ⊢ ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒)) |
5 | 4 | ralbii2 3160 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 = wceq 1653 ∈ wcel 2157 ∀wral 3090 |
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 ax-6 2072 ax-7 2107 ax-9 2166 ax-ext 2778 |
This theorem depends on definitions: df-bi 199 df-an 386 df-ex 1876 df-cleq 2793 df-clel 2796 df-ral 3095 |
This theorem is referenced by: wfrlem5 7659 ply1coe 19987 ordtbaslem 21320 iscusp2 22433 isrgr 26808 frrlem5 32296 elghomOLD 34172 iscrngo2 34282 tendoset 36779 comptiunov2i 38776 |
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