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Mirrors > Home > MPE Home > Th. List > equsexv | Structured version Visualization version GIF version |
Description: An equivalence related to implicit substitution. Version of equsex 2440 with a disjoint variable condition, which does not require ax-13 2390. See equsexvw 2011 for a version with two disjoint variable conditions requiring fewer axioms. See also the dual form equsalv 2268. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) |
Ref | Expression |
---|---|
equsalv.nf | ⊢ Ⅎ𝑥𝜓 |
equsalv.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
equsexv | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsalv.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
2 | 1 | pm5.32i 577 | . . 3 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) ↔ (𝑥 = 𝑦 ∧ 𝜓)) |
3 | 2 | exbii 1848 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜓)) |
4 | ax6ev 1972 | . . 3 ⊢ ∃𝑥 𝑥 = 𝑦 | |
5 | equsalv.nf | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
6 | 5 | 19.41 2237 | . . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ (∃𝑥 𝑥 = 𝑦 ∧ 𝜓)) |
7 | 4, 6 | mpbiran 707 | . 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜓) ↔ 𝜓) |
8 | 3, 7 | bitri 277 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∃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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-nf 1785 |
This theorem is referenced by: sb5 2276 sb56OLD 2278 equsexhv 2300 cleljustALT2 2383 sb10f 2571 dprd2d2 19168 poimirlem25 34919 |
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