![]() |
Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
|
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 3111 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
4 | sp 2180 | . . . 4 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) → (𝑥 ∈ 𝐴 → 𝜓)) | |
5 | 4 | impcom 411 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) → 𝜓) |
6 | 3, 5 | sylan2b 596 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝜓) → 𝜓) |
7 | 1, 2, 6 | syl2anc 587 | 1 ⊢ (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1536 ∈ wcel 2111 ∀wral 3106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-ral 3111 |
This theorem is referenced by: bnj1379 32212 bnj1121 32367 bnj1279 32400 bnj1286 32401 bnj1296 32403 bnj1421 32424 bnj1489 32438 bnj1501 32449 bnj1523 32453 |
Copyright terms: Public domain | W3C validator |