Theorem elabgf1 15648

Description: One implication of elabgf 2914. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
elabgf1.nf1	⊢ Ⅎ𝑥𝐴
elabgf1.nf2	⊢ Ⅎ𝑥𝜓
elabgf1.1	⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))

Assertion

Ref	Expression
elabgf1	⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)

Proof of Theorem elabgf1

Step	Hyp	Ref	Expression
1		elabgf1.nf1	. . 3 ⊢ Ⅎ𝑥𝐴
2		elabgf1.nf2	. . 3 ⊢ Ⅎ𝑥𝜓
3	1, 2	elabgft1 15647	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))
4		elabgf1.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
5	3, 4	mpg 1473	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)

Syntax hints:   → wi 4   = wceq 1372  Ⅎwnf 1482   ∈ wcel 2175  {cab 2190  Ⅎwnfc 2334

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773

This theorem is referenced by:  elabf1  15650