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Mirrors > Home > MPE Home > Th. List > Mathboxes > dveel2ALT | Structured version Visualization version GIF version |
Description: Alternate proof of dveel2 2462 using ax-c16 36529 instead of ax-5 1917. (Contributed by NM, 10-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dveel2ALT | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax5el 36574 | . 2 ⊢ (𝑧 ∈ 𝑤 → ∀𝑥 𝑧 ∈ 𝑤) | |
2 | ax5el 36574 | . 2 ⊢ (𝑧 ∈ 𝑦 → ∀𝑤 𝑧 ∈ 𝑦) | |
3 | elequ2 2129 | . 2 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦)) | |
4 | 1, 2, 3 | dvelimh 2450 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 ∈ 𝑦 → ∀𝑥 𝑧 ∈ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-13 2372 ax-c14 36528 ax-c16 36529 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 |
This theorem is referenced by: (None) |
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