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Mirrors > Home > ILE Home > Th. List > ralbii2 | GIF version |
Description: Inference adding different restricted universal quantifiers to each side of an equivalence. (Contributed by NM, 15-Aug-2005.) |
Ref | Expression |
---|---|
ralbii2.1 | ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐵 → 𝜓)) |
Ref | Expression |
---|---|
ralbii2 | ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralbii2.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) ↔ (𝑥 ∈ 𝐵 → 𝜓)) | |
2 | 1 | albii 1463 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓)) |
3 | df-ral 2453 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
4 | df-ral 2453 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓)) | |
5 | 2, 3, 4 | 3bitr4i 211 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐵 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1346 ∈ wcel 2141 ∀wral 2448 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-gen 1442 |
This theorem depends on definitions: df-bi 116 df-ral 2453 |
This theorem is referenced by: raleqbii 2482 ralbiia 2484 ralrab 2891 raldifb 3267 raluz2 9525 ralrp 9619 isprm4 12060 |
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