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Mirrors > Home > ILE Home > Th. List > dvelimfALT2 | GIF version |
Description: Proof of dvelimf 1940 using dveeq2 1744 (shown as the last hypothesis) instead of ax-12 1448. This shows that ax-12 1448 could be replaced by dveeq2 1744 (the last hypothesis). (Contributed by Andrew Salmon, 21-Jul-2011.) |
Ref | Expression |
---|---|
dvelimfALT2.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
dvelimfALT2.2 | ⊢ (𝜓 → ∀𝑧𝜓) |
dvelimfALT2.3 | ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) |
dvelimfALT2.4 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) |
Ref | Expression |
---|---|
dvelimfALT2 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-17 1465 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦) | |
2 | hbn1 1588 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑥 ¬ ∀𝑥 𝑥 = 𝑦) | |
3 | dvelimfALT2.4 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑧 = 𝑦 → ∀𝑥 𝑧 = 𝑦)) | |
4 | dvelimfALT2.1 | . . . . 5 ⊢ (𝜑 → ∀𝑥𝜑) | |
5 | 4 | a1i 9 | . . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑)) |
6 | 2, 3, 5 | hbimd 1511 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ((𝑧 = 𝑦 → 𝜑) → ∀𝑥(𝑧 = 𝑦 → 𝜑))) |
7 | 1, 6 | hbald 1426 | . 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (∀𝑧(𝑧 = 𝑦 → 𝜑) → ∀𝑥∀𝑧(𝑧 = 𝑦 → 𝜑))) |
8 | dvelimfALT2.2 | . . 3 ⊢ (𝜓 → ∀𝑧𝜓) | |
9 | dvelimfALT2.3 | . . 3 ⊢ (𝑧 = 𝑦 → (𝜑 ↔ 𝜓)) | |
10 | 8, 9 | equsalh 1662 | . 2 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ 𝜓) |
11 | 10 | albii 1405 | . 2 ⊢ (∀𝑥∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ ∀𝑥𝜓) |
12 | 7, 10, 11 | 3imtr3g 203 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∀wal 1288 = wceq 1290 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-fal 1296 |
This theorem is referenced by: (None) |
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