Theorem ss2abdvOLD 3995

Description: Obsolete version of ss2abdv 3993 as of 28-Jun-2024. (Contributed by NM, 29-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ss2abdvOLD.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
ss2abdvOLD	⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒})

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem ss2abdvOLD

Step	Hyp	Ref	Expression
1		ss2abdvOLD.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	alrimiv 1931	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
3		ss2ab 3989	. 2 ⊢ ({𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒} ↔ ∀𝑥(𝜓 → 𝜒))
4	2, 3	sylibr 233	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒})

Syntax hints:   → wi 4  ∀wal 1537  {cab 2715   ⊆ wss 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-v 3424  df-in 3890  df-ss 3900

This theorem is referenced by: (None)