Theorem ssopab2i 4928

Description: Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)

Hypothesis

Ref	Expression
ssopab2i.1	⊢ (𝜑 → 𝜓)

Assertion

Ref	Expression
ssopab2i	⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓}

Proof of Theorem ssopab2i

Step	Hyp	Ref	Expression
1		ssopab2 4926	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
2		ssopab2i.1	. . 3 ⊢ (𝜑 → 𝜓)
3	2	ax-gen 1713	. 2 ⊢ ∀𝑦(𝜑 → 𝜓)
4	1, 3	mpg 1715	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓}

Syntax hints:   → wi 4  ∀wal 1473   ⊆ wss 3540  {copab 4642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-in 3547  df-ss 3554  df-opab 4644

This theorem is referenced by:  elopabran  4938  brab2a  5091  opabssxp  5116  funopab4  5839  ssoprab2i  6647  cardf2  8652  dfac3  8827  axdc2lem  9153  fpwwe2lem1  9332  canthwe  9352  trclublem  13582  fullfunc  16389  fthfunc  16390  isfull  16393  isfth  16397  ipoval  16977  ipolerval  16979  eqgfval  17465  2ndcctbss  21068  iscgrg  25207  ishpg  25451  nvss  26832  ajfval  27048  afsval  30002  cvmlift2lem12  30550  dicval  35483  areaquad  36821  relopabVD  38159  pthsfval  40927  spthsfval  40928  crctS  40994  cyclS  40995  eupths  41367