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Mirrors > Home > ILE Home > Th. List > 2exbidv | GIF version |
Description: Formula-building rule for 2 existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.) |
Ref | Expression |
---|---|
2albidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
2exbidv | ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑥∃𝑦𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2albidv.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | exbidv 1818 | . 2 ⊢ (𝜑 → (∃𝑦𝜓 ↔ ∃𝑦𝜒)) |
3 | 2 | exbidv 1818 | 1 ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑥∃𝑦𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∃wex 1485 |
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 ax-ie1 1486 ax-ie2 1487 ax-4 1503 ax-17 1519 ax-ial 1527 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: 3exbidv 1862 4exbidv 1863 cbvex4v 1923 ceqsex3v 2772 ceqsex4v 2773 copsexg 4229 euotd 4239 elopab 4243 elxpi 4627 relop 4761 cbvoprab3 5929 ov6g 5990 th3qlem1 6615 ltresr 7801 fisumcom2 11401 fprodcom2fi 11589 |
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