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Mirrors > Home > MPE Home > Th. List > 2ralbiim | Structured version Visualization version GIF version |
Description: Split a biconditional and distribute two restricted universal quantifiers, analogous to 2albiim 1897 and ralbiim 3101. (Contributed by Alexander van der Vekens, 2-Jul-2017.) |
Ref | Expression |
---|---|
2ralbiim | ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ↔ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralbiim 3101 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 (𝜑 ↔ 𝜓) ↔ (∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ∧ ∀𝑦 ∈ 𝐵 (𝜓 → 𝜑))) | |
2 | 1 | ralbii 3093 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ↔ 𝜓) ↔ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ∧ ∀𝑦 ∈ 𝐵 (𝜓 → 𝜑))) |
3 | r19.26 3097 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ∧ ∀𝑦 ∈ 𝐵 (𝜓 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝜑))) | |
4 | 2, 3 | bitri 274 | 1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 ↔ 𝜓) ↔ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝜓) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wral 3066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ral 3071 |
This theorem is referenced by: thincciso 46309 |
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