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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 1893 | . . 3 ⊢ (∀𝑦(𝜑 ↔ 𝜓) ↔ (∀𝑦(𝜑 → 𝜓) ∧ ∀𝑦(𝜓 → 𝜑))) | |
2 | 1 | albii 1822 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) ↔ ∀𝑥(∀𝑦(𝜑 → 𝜓) ∧ ∀𝑦(𝜓 → 𝜑))) |
3 | 19.26 1874 | . 2 ⊢ (∀𝑥(∀𝑦(𝜑 → 𝜓) ∧ ∀𝑦(𝜓 → 𝜑)) ↔ (∀𝑥∀𝑦(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦(𝜓 → 𝜑))) | |
4 | 2, 3 | bitri 275 | 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 397 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 df-an 398 |
This theorem is referenced by: sbnf2 2355 2eu6 2653 eqopab2bw 5549 eqopab2b 5553 eqrel 5785 eqrelrel 5798 eqoprab2bw 7479 eqoprab2b 7480 eqrelrd2 31845 eqrel2 37168 relcnveq2 37192 elrelscnveq2 37363 pm14.123a 43184 |
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