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Mirrors > Home > MPE Home > Th. List > Mathboxes > alrim3con13v | Structured version Visualization version GIF version |
Description: Closed form of alrimi 2069 with 2 additional conjuncts having no occurrences of the quantifying variable. This proof is 19.21a3con13vVD 38109 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
alrim3con13v | ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1054 | . . . . 5 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜓) | |
2 | 1 | a1i 11 | . . . 4 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜓)) |
3 | ax-5 1827 | . . . 4 ⊢ (𝜓 → ∀𝑥𝜓) | |
4 | 2, 3 | syl6 34 | . . 3 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥𝜓)) |
5 | simp2 1055 | . . . 4 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜑) | |
6 | 5 | imim1i 61 | . . 3 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥𝜑)) |
7 | simp3 1056 | . . . . 5 ⊢ ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜒) | |
8 | 7 | a1i 11 | . . . 4 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → 𝜒)) |
9 | ax-5 1827 | . . . 4 ⊢ (𝜒 → ∀𝑥𝜒) | |
10 | 8, 9 | syl6 34 | . . 3 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥𝜒)) |
11 | 4, 6, 10 | 3jcad 1236 | . 2 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → (∀𝑥𝜓 ∧ ∀𝑥𝜑 ∧ ∀𝑥𝜒))) |
12 | 19.26-3an 1788 | . 2 ⊢ (∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒) ↔ (∀𝑥𝜓 ∧ ∀𝑥𝜑 ∧ ∀𝑥𝜒)) | |
13 | 11, 12 | syl6ibr 241 | 1 ⊢ ((𝜑 → ∀𝑥𝜑) → ((𝜓 ∧ 𝜑 ∧ 𝜒) → ∀𝑥(𝜓 ∧ 𝜑 ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 ∀wal 1473 |
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 |
This theorem depends on definitions: df-bi 196 df-an 385 df-3an 1033 |
This theorem is referenced by: tratrbVD 38119 |
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