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Mirrors > Home > MPE Home > Th. List > equsex | Structured version Visualization version GIF version |
Description: An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2367. See equsexvw 2001 and equsexv 2255 for versions with disjoint variable conditions proved from fewer axioms. See also the dual form equsal 2412. See equsexALT 2414 for an alternate proof. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Feb-2018.) (New usage is discouraged.) |
Ref | Expression |
---|---|
equsal.1 | ⊢ Ⅎ𝑥𝜓 |
equsal.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
equsex | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsal.1 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
2 | equsal.2 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
3 | 2 | biimpa 476 | . . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜓) |
4 | 1, 3 | exlimi 2206 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜓) |
5 | 1, 2 | equsal 2412 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
6 | equs4 2411 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
7 | 5, 6 | sylbir 234 | . 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
8 | 4, 7 | impbii 208 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1532 ∃wex 1774 Ⅎwnf 1778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-12 2167 ax-13 2367 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-nf 1779 |
This theorem is referenced by: equsexh 2416 sb5rf 2462 |
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