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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 2390. See equsexvw 2011 and equsexv 2269 for versions with disjoint variable conditions proved from fewer axioms. See also the dual form equsal 2439. See equsexALT 2441 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 479 | . . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → 𝜓) |
4 | 1, 3 | exlimi 2217 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → 𝜓) |
5 | 1, 2 | equsal 2439 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓) |
6 | equs4 2438 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) | |
7 | 5, 6 | sylbir 237 | . 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
8 | 4, 7 | impbii 211 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 ∃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 1911 ax-6 1970 ax-7 2015 ax-12 2177 ax-13 2390 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 |
This theorem is referenced by: equsexh 2443 sb5rf 2490 |
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