Theorem ss2abdvOLD 4054

Description: Obsolete version of ss2abdv 4052 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 1922	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
3		ss2ab 4048	. 2 ⊢ ({𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒} ↔ ∀𝑥(𝜓 → 𝜒))
4	2, 3	sylibr 233	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒})

Syntax hints:   → wi 4  ∀wal 1531  {cab 2701   ⊆ wss 3940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-v 3468  df-in 3947  df-ss 3957

This theorem is referenced by: (None)