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Mathbox for Jonathan Ben-Naim |
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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 2946 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
4 | sp 2091 | . . . 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 694 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1521 ∈ wcel 2030 ∀wral 2941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-12 2087 |
This theorem depends on definitions: df-bi 197 df-an 385 df-ex 1745 df-ral 2946 |
This theorem is referenced by: bnj1379 31027 bnj1121 31179 bnj1279 31212 bnj1286 31213 bnj1296 31215 bnj1421 31236 bnj1450 31244 bnj1489 31250 bnj1501 31261 bnj1523 31265 |
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