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Mirrors > Home > MPE Home > Th. List > ss2abdvALT | Structured version Visualization version GIF version |
Description: Alternate proof of ss2abdv 4021. Shorter, but requiring ax-8 2109. (Contributed by Steven Nguyen, 28-Jun-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ss2abdvALT.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
ss2abdvALT | ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ss2abdvALT.1 | . . . 4 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | sbimdv 2082 | . . 3 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)) |
3 | df-clab 2711 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) | |
4 | df-clab 2711 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜒} ↔ [𝑦 / 𝑥]𝜒) | |
5 | 2, 3, 4 | 3imtr4g 296 | . 2 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∣ 𝜓} → 𝑦 ∈ {𝑥 ∣ 𝜒})) |
6 | 5 | ssrdv 3951 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 2068 ∈ wcel 2107 {cab 2710 ⊆ wss 3911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-in 3918 df-ss 3928 |
This theorem is referenced by: (None) |
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