![]() |
Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1441 | 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 |
---|---|
bnj1441.1 | ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴) |
bnj1441.2 | ⊢ (𝜑 → ∀𝑦𝜑) |
Ref | Expression |
---|---|
bnj1441 | ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → ∀𝑦 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3126 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | bnj1441.1 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 𝑥 ∈ 𝐴) | |
3 | bnj1441.2 | . . . 4 ⊢ (𝜑 → ∀𝑦𝜑) | |
4 | 2, 3 | hban 2334 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → ∀𝑦(𝑥 ∈ 𝐴 ∧ 𝜑)) |
5 | 4 | hbab 2816 | . 2 ⊢ (𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} → ∀𝑦 𝑧 ∈ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
6 | 1, 5 | hbxfreq 2935 | 1 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑} → ∀𝑦 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∀wal 1656 ∈ wcel 2166 {cab 2811 {crab 3121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2812 df-cleq 2818 df-clel 2821 df-rab 3126 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |