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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 1897 | . . 3 ⊢ (∀𝑦(𝜑 ↔ 𝜓) ↔ (∀𝑦(𝜑 → 𝜓) ∧ ∀𝑦(𝜓 → 𝜑))) | |
2 | 1 | albii 1827 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) ↔ ∀𝑥(∀𝑦(𝜑 → 𝜓) ∧ ∀𝑦(𝜓 → 𝜑))) |
3 | 19.26 1878 | . 2 ⊢ (∀𝑥(∀𝑦(𝜑 → 𝜓) ∧ ∀𝑦(𝜓 → 𝜑)) ↔ (∀𝑥∀𝑦(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦(𝜓 → 𝜑))) | |
4 | 2, 3 | bitri 278 | 1 ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) ↔ (∀𝑥∀𝑦(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦(𝜓 → 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 |
This theorem depends on definitions: df-bi 210 df-an 400 |
This theorem is referenced by: sbnf2 2358 2eu6 2658 eqopab2bw 5444 eqopab2b 5448 eqrel 5670 eqrelrel 5682 eqoprab2bw 7300 eqoprab2b 7301 eqrelrd2 30702 eqrel2 36202 relcnveq2 36225 elrelscnveq2 36378 pm14.123a 41749 |
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