Theorem ssopab2i 4415

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 4413	. 2 ⊢ (∀𝑥∀𝑦(𝜑 → 𝜓) → {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓})
2		ssopab2i.1	. . 3 ⊢ (𝜑 → 𝜓)
3	2	ax-gen 1502	. 2 ⊢ ∀𝑦(𝜑 → 𝜓)
4	1, 3	mpg 1504	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓}

Syntax hints:   → wi 4  ∀wal 1400   ⊆ wss 3220  {copab 4186

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  df-opab 4188

This theorem is referenced by:  elopabran  4421  brab2a  4823  opabssxp  4844  relopabiv  4898  funopab4  5409  ssoprab2i  6167  npsspw  7828  eqgfval  14002  trlsex  16542  eupthsg  16600