![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 3exbidv | Structured version Visualization version GIF version |
Description: Formula-building rule for three existential quantifiers (deduction form). (Contributed by NM, 1-May-1995.) |
Ref | Expression |
---|---|
3exbidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
3exbidv | ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧𝜓 ↔ ∃𝑥∃𝑦∃𝑧𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3exbidv.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | exbidv 1922 | . 2 ⊢ (𝜑 → (∃𝑧𝜓 ↔ ∃𝑧𝜒)) |
3 | 2 | 2exbidv 1925 | 1 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧𝜓 ↔ ∃𝑥∃𝑦∃𝑧𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∃wex 1779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 |
This theorem depends on definitions: df-bi 206 df-ex 1780 |
This theorem is referenced by: ceqsex6v 3532 euotd 5512 oprabidw 7442 oprabid 7443 0mpo0 7494 eloprabga 7518 eloprabgaOLD 7519 eloprabi 8051 bnj981 34259 fundcmpsurbijinj 46376 |
Copyright terms: Public domain | W3C validator |