![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > equsexvw | Structured version Visualization version GIF version |
Description: Version of equsexv 2265 with a dv condition, and of equsex 2447 with two dv conditions, which requires fewer axioms. See also the dual form equsalvw 2089. (Contributed by BJ, 31-May-2019.) |
Ref | Expression |
---|---|
equsalvw.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
equsexvw | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsalvw.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | pm5.32i 564 | . . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) ↔ (𝑥 = 𝑦 ∧ 𝜓)) |
3 | 2 | exbii 1924 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) |
4 | ax6ev 2059 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
5 | 19.41v 2029 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ (∃𝑥 𝑥 = 𝑦 ∧ 𝜓)) | |
6 | 4, 5 | mpbiran 688 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜓) |
7 | 3, 6 | bitri 264 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∃wex 1852 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 |
This theorem depends on definitions: df-bi 197 df-an 383 df-ex 1853 |
This theorem is referenced by: equvinv 2115 cleljust 2153 sbhypf 3405 axsep 4914 dfid3 5158 opeliunxp 5310 imai 5619 coi1 5795 elfuns 32359 bj-axsep 33129 |
Copyright terms: Public domain | W3C validator |