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Mirrors > Home > MPE Home > Th. List > cleljustALT2 | Structured version Visualization version GIF version |
Description: Alternate proof of cleljust 1985. Compared with cleljustALT 2173, it uses nfv 1830 followed by equsexv 2095 instead of ax-5 1827 followed by equsexhv 2135, so it uses the idiom Ⅎ𝑥𝜑 instead of 𝜑 → ∀𝑥𝜑 to express non-freeness. This style is generally preferred for later theorems. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.) (Revised by BJ, 29-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cleljustALT2 | ⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1830 | . . 3 ⊢ Ⅎ𝑧 𝑥 ∈ 𝑦 | |
2 | elequ1 1984 | . . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦)) | |
3 | 1, 2 | equsexv 2095 | . 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) ↔ 𝑥 ∈ 𝑦) |
4 | 3 | bicomi 213 | 1 ⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-nf 1701 |
This theorem is referenced by: (None) |
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