![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > equsexALT | Structured version Visualization version GIF version |
Description: Alternate proof of equsex 2416. This proves the result directly, instead of as a corollary of equsal 2415 via equs4 2414. Note in particular that only existential quantifiers appear in the proof and that the only step requiring ax-13 2370 is ax6e 2381. This proof mimics that of equsal 2415 (in particular, note that pm5.32i 574, exbii 1849, 19.41 2227, mpbiran 706 correspond respectively to pm5.74i 270, albii 1820, 19.23 2203, a1bi 361). (Contributed by BJ, 20-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
equsal.1 | ⊢ Ⅎ𝑥𝜓 |
equsal.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
equsexALT | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsal.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | pm5.32i 574 | . . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) ↔ (𝑥 = 𝑦 ∧ 𝜓)) |
3 | 2 | exbii 1849 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) |
4 | ax6e 2381 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
5 | equsal.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
6 | 5 | 19.41 2227 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ (∃𝑥 𝑥 = 𝑦 ∧ 𝜓)) |
7 | 4, 6 | mpbiran 706 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜓) |
8 | 3, 7 | bitri 274 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∃wex 1780 Ⅎwnf 1784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-12 2170 ax-13 2370 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1781 df-nf 1785 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |