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Mirrors > Home > MPE Home > Th. List > 2albiim | Structured version Visualization version GIF version |
Description: Split a biconditional and distribute two quantifiers. (Contributed by NM, 3-Feb-2005.) |
Ref | Expression |
---|---|
2albiim | ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) ↔ (∀𝑥∀𝑦(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦(𝜓 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | albiim 1890 | . . 3 ⊢ (∀𝑦(𝜑 ↔ 𝜓) ↔ (∀𝑦(𝜑 → 𝜓) ∧ ∀𝑦(𝜓 → 𝜑))) | |
2 | 1 | albii 1819 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) ↔ ∀𝑥(∀𝑦(𝜑 → 𝜓) ∧ ∀𝑦(𝜓 → 𝜑))) |
3 | 19.26 1871 | . 2 ⊢ (∀𝑥(∀𝑦(𝜑 → 𝜓) ∧ ∀𝑦(𝜓 → 𝜑)) ↔ (∀𝑥∀𝑦(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦(𝜓 → 𝜑))) | |
4 | 2, 3 | bitri 274 | 1 ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) ↔ (∀𝑥∀𝑦(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦(𝜓 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∀wal 1537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
This theorem depends on definitions: df-bi 206 df-an 395 |
This theorem is referenced by: sbnf2 2352 2eu6 2650 eqopab2bw 5547 eqopab2b 5551 eqrel 5783 eqrelrel 5796 eqoprab2bw 7481 eqoprab2b 7482 eqrelrd2 32112 eqrel2 37471 relcnveq2 37495 elrelscnveq2 37666 pm14.123a 43486 |
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