Theorem ss2abdvALT 3968

Description: Alternate proof of ss2abdv 3967. Shorter, but requiring ax-8 2112. (Contributed by Steven Nguyen, 28-Jun-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ss2abdvALT.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
ss2abdvALT	⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒})

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem ss2abdvALT

Dummy variable 𝑦 is distinct from all other variables.

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 2713	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} ↔ [𝑦 / 𝑥]𝜓)
4		df-clab 2713	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜒} ↔ [𝑦 / 𝑥]𝜒)
5	2, 3, 4	3imtr4g 299	. 2 ⊢ (𝜑 → (𝑦 ∈ {𝑥 ∣ 𝜓} → 𝑦 ∈ {𝑥 ∣ 𝜒}))
6	5	ssrdv 3897	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒})

Syntax hints:   → wi 4  [wsb 2070   ∈ wcel 2110  {cab 2712   ⊆ wss 3857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2706

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812  df-v 3403  df-in 3864  df-ss 3874

This theorem is referenced by: (None)