Theorem ss2abdv 3321

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 1927	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → 𝜒))
3		ss2ab 3316	. 2 ⊢ ({𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒} ↔ ∀𝑥(𝜓 → 𝜒))
4	2, 3	sylibr 134	1 ⊢ (𝜑 → {𝑥 ∣ 𝜓} ⊆ {𝑥 ∣ 𝜒})

Syntax hints:   → wi 4  ∀wal 1400  {cab 2224   ⊆ wss 3220

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-in 3226  df-ss 3233

This theorem is referenced by:  ssopab2  4416  iotass  5353  imadif  5459  imain  5461  opabbrex  6125  ssoprab2  6137  ressuppss  6487  tfr1onlemssrecs  6603  tfrcllemssrecs  6616  ss2ixp  6986  ptex  13598  plyss  15765