Theorem ssopab2i 4194

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 4192	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
2		ssopab2i.1	. . 3 ⊢ (𝜑 → 𝜓)
3	2	ax-gen 1425	. 2 ⊢ ∀𝑦(𝜑 → 𝜓)
4	1, 3	mpg 1427	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓}

Syntax hints:   → wi 4  ∀wal 1329   ⊆ wss 3066  {copab 3983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-in 3072  df-ss 3079  df-opab 3985

This theorem is referenced by:  brab2a  4587  opabssxp  4608  funopab4  5155  ssoprab2i  5853  npsspw  7272