![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ssabdv | Structured version Visualization version GIF version |
Description: Deduction of abstraction subclass from implication. (Contributed by SN, 22-Dec-2024.) |
Ref | Expression |
---|---|
ssabdv.1 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) |
Ref | Expression |
---|---|
ssabdv | ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∣ 𝜓}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abid1 2866 | . 2 ⊢ 𝐴 = {𝑥 ∣ 𝑥 ∈ 𝐴} | |
2 | ssabdv.1 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) | |
3 | 2 | ss2abdv 4058 | . 2 ⊢ (𝜑 → {𝑥 ∣ 𝑥 ∈ 𝐴} ⊆ {𝑥 ∣ 𝜓}) |
4 | 1, 3 | eqsstrid 4028 | 1 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∣ 𝜓}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 {cab 2705 ⊆ wss 3947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3473 df-in 3954 df-ss 3964 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |