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Mirrors > Home > MPE Home > Th. List > equsexhv | Structured version Visualization version GIF version |
Description: An equivalence related to implicit substitution. Version of equsexh 2420 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 5-Aug-1993.) (Revised by BJ, 31-May-2019.) |
Ref | Expression |
---|---|
equsalhw.1 | ⊢ (𝜓 → ∀𝑥𝜓) |
equsalhw.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
equsexhv | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equsalhw.1 | . . 3 ⊢ (𝜓 → ∀𝑥𝜓) | |
2 | 1 | nf5i 2143 | . 2 ⊢ Ⅎ𝑥𝜓 |
3 | equsalhw.2 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | equsexv 2260 | 1 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ↔ 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-10 2138 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-nf 1787 |
This theorem is referenced by: cleljustALT 2361 |
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