Theorem elabf1 16377

Description: One implication of elabf 2949. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
elabf1.nf	⊢ Ⅎ𝑥𝜓
elabf1.1	⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))

Assertion

Ref	Expression
elabf1	⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem elabf1

Step	Hyp	Ref	Expression
1		nfcv 2374	. 2 ⊢ Ⅎ𝑥𝐴
2		elabf1.nf	. 2 ⊢ Ⅎ𝑥𝜓
3		elabf1.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
4	1, 2, 3	elabgf1 16375	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)

Syntax hints:   → wi 4   = wceq 1397  Ⅎwnf 1508   ∈ wcel 2202  {cab 2217

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804

This theorem is referenced by:  elab1  16379  bj-bdfindis  16542