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Mirrors > Home > MPE Home > Th. List > equsexALT | Structured version Visualization version GIF version |
Description: Alternate proof of equsex 2421. This proves the result directly, instead of as a corollary of equsal 2420 via equs4 2419. Note in particular that only existential quantifiers appear in the proof and that the only step requiring ax-13 2375 is ax6e 2386. This proof mimics that of equsal 2420 (in particular, note that pm5.32i 574, exbii 1845, 19.41 2233, mpbiran 709 correspond respectively to pm5.74i 271, albii 1816, 19.23 2209, a1bi 362). (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 1845 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) |
4 | ax6e 2386 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
5 | equsal.1 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
6 | 5 | 19.41 2233 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ (∃𝑥 𝑥 = 𝑦 ∧ 𝜓)) |
7 | 4, 6 | mpbiran 709 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜓) |
8 | 3, 7 | bitri 275 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∃wex 1776 Ⅎwnf 1780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-12 2175 ax-13 2375 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-nf 1781 |
This theorem is referenced by: (None) |
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