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Mirrors > Home > ILE Home > Th. List > 3exbii | GIF version |
Description: Inference adding 3 existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.) |
Ref | Expression |
---|---|
exbii.1 | ⊢ (𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
3exbii | ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exbii.1 | . . 3 ⊢ (𝜑 ↔ 𝜓) | |
2 | 1 | exbii 1537 | . 2 ⊢ (∃𝑧𝜑 ↔ ∃𝑧𝜓) |
3 | 2 | 2exbii 1538 | 1 ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 ∃wex 1422 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1377 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-4 1441 ax-ial 1468 |
This theorem depends on definitions: df-bi 115 |
This theorem is referenced by: eeeanv 1851 ceqsex6v 2651 oprabid 5588 dfoprab2 5603 dftpos3 5931 xpassen 6395 |
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