Theorem ss2abdv 3301

Description: Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)

Hypothesis

Ref	Expression
ss2abdv.1	⊢ (𝜑 → (𝜓 → 𝜒))

Assertion

Ref	Expression
ss2abdv	⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒})

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem ss2abdv

Step	Hyp	Ref	Expression
1		ss2abdv.1	. . 3 ⊢ (𝜑 → (𝜓 → 𝜒))
2	1	alrimiv 1922	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
3		ss2ab 3296	. 2 ⊢ ({𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒} ↔ ∀𝑥(𝜓 → 𝜒))
4	2, 3	sylibr 134	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒})

Syntax hints:   → wi 4  ∀wal 1396  {cab 2217   ⊆ wss 3201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-in 3207  df-ss 3214

This theorem is referenced by:  ssopab2  4376  iotass  5311  imadif  5417  imain  5419  opabbrex  6075  ssoprab2  6087  ressuppss  6432  tfr1onlemssrecs  6548  tfrcllemssrecs  6561  ss2ixp  6923  ptex  13408  plyss  15529