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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 2680 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) |
3 | raleqbii.2 | . . 3 ⊢ (𝜓 ↔ 𝜒) | |
4 | 2, 3 | imbi12i 339 | . 2 ⊢ ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒)) |
5 | 4 | ralbii2 2961 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∀wral 2896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-cleq 2603 df-clel 2606 df-ral 2901 |
This theorem is referenced by: wfrlem5 7306 ply1coe 19487 ordtbaslem 20802 iscusp2 21916 frrlem5 31028 elghomOLD 32856 iscrngo2 32966 tendoset 35065 comptiunov2i 37017 isrgr 40759 |
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