Theorem elab2a 11026

Description: One implication of elab 2748. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
elab2a.s	⊢ 𝐴 ∈ V
elab2a.1	⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))

Assertion

Ref	Expression
elab2a	⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})

Distinct variable groups:   𝜓,𝑥   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elab2a

Step	Hyp	Ref	Expression
1		nfv 1462	. 2 ⊢ Ⅎ𝑥𝜓
2		elab2a.s	. 2 ⊢ 𝐴 ∈ V
3		elab2a.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))
4	1, 2, 3	elabf2 11024	1 ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1285   ∈ wcel 1434  {cab 2069  Vcvv 2612

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614

This theorem is referenced by: (None)