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Mirrors > Home > MPE Home > Th. List > ss2abdvALT | Structured version Visualization version GIF version |
Description: Alternate proof of ss2abdv 3964. Shorter, but requiring ax-8 2114. (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 2084 | . . 3 ⊢ (𝜑 → ([𝑦 / 𝑥]𝜓 → [𝑦 / 𝑥]𝜒)) |
3 | df-clab 2737 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓) | |
4 | df-clab 2737 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜒} ↔ [𝑦 / 𝑥]𝜒) | |
5 | 2, 3, 4 | 3imtr4g 300 | . 2 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∣ 𝜓} → 𝑦 ∈ {𝑥 ∣ 𝜒})) |
6 | 5 | ssrdv 3894 | 1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 [wsb 2070 ∈ wcel 2112 {cab 2736 ⊆ wss 3854 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2730 |
This theorem depends on definitions: df-bi 210 df-an 401 df-tru 1542 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-v 3409 df-in 3861 df-ss 3871 |
This theorem is referenced by: (None) |
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