Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2906 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵) |
3 | raleqbii.2 | . . 3 ⊢ (𝜓 ↔ 𝜒) | |
4 | 2, 3 | imbi12i 353 | . 2 ⊢ ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑥 ∈ 𝐵 → 𝜒)) |
5 | 4 | ralbii2 3165 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∀wral 3140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-cleq 2816 df-clel 2895 df-ral 3145 |
This theorem is referenced by: wfrlem5 7961 ply1coe 20466 ordtbaslem 21798 iscusp2 22913 isrgr 27343 fprlem1 33139 frrlem15 33144 elghomOLD 35167 iscrngo2 35277 tendoset 37897 comptiunov2i 40058 |
Copyright terms: Public domain | W3C validator |