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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 1569 | . 2 ⊢ (∃𝑧𝜑 ↔ ∃𝑧𝜓) |
3 | 2 | 2exbii 1570 | 1 ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∃wex 1453 |
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 1408 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-4 1472 ax-ial 1499 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: eeeanv 1885 ceqsex6v 2704 oprabid 5771 dfoprab2 5786 dftpos3 6127 xpassen 6692 |
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