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Mirrors > Home > ILE Home > Th. List > dvelimfALT2 | GIF version |
Description: Proof of dvelimf 2027 using dveeq2 1826 (shown as the last hypothesis) instead of ax12 1523. This shows that ax12 1523 could be replaced by dveeq2 1826 (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 1537 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ∀𝑧 ¬ ∀𝑥 𝑥 = 𝑦) | |
2 | hbn1 1663 | . . . 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 1584 | . . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ((𝑧 = 𝑦 → 𝜑) → ∀𝑥(𝑧 = 𝑦 → 𝜑))) |
7 | 1, 6 | hbald 1502 | . 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 1737 | . 2 ⊢ (∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ 𝜓) |
11 | 10 | albii 1481 | . 2 ⊢ (∀𝑥∀𝑧(𝑧 = 𝑦 → 𝜑) ↔ ∀𝑥𝜓) |
12 | 7, 10, 11 | 3imtr3g 204 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝜓 → ∀𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 ∀wal 1362 = wceq 1364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-fal 1370 |
This theorem is referenced by: (None) |
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