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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1294 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1294.1 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) |
bnj1294.2 | ⊢ (𝜑 → 𝑥 ∈ 𝐴) |
Ref | Expression |
---|---|
bnj1294 | ⊢ (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1294.2 | . 2 ⊢ (𝜑 → 𝑥 ∈ 𝐴) | |
2 | bnj1294.1 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) | |
3 | df-ral 2901 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
4 | sp 2041 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) → (𝑥 ∈ 𝐴 → 𝜓)) | |
5 | 4 | impcom 445 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) → 𝜓) |
6 | 3, 5 | sylan2b 491 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜓) → 𝜓) |
7 | 1, 2, 6 | syl2anc 691 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 ∈ wcel 1977 ∀wral 2896 |
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-12 2034 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-ral 2901 |
This theorem is referenced by: bnj1379 30155 bnj1121 30307 bnj1279 30340 bnj1286 30341 bnj1296 30343 bnj1421 30364 bnj1450 30372 bnj1489 30378 bnj1501 30389 bnj1523 30393 |
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