Theorem elabf2 11024

Description: One implication of elabf 2747. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
elabf2.nf	⊢ Ⅎ𝑥𝜓
elabf2.s	⊢ 𝐴 ∈ V
elabf2.1	⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))

Assertion

Ref	Expression
elabf2	⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem elabf2

Step	Hyp	Ref	Expression
1		elabf2.s	. 2 ⊢ 𝐴 ∈ V
2		nfcv 2223	. . 3 ⊢ Ⅎ𝑥𝐴
3		elabf2.nf	. . 3 ⊢ Ⅎ𝑥𝜓
4		elabf2.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))
5	2, 3, 4	elabgf2 11022	. 2 ⊢ (𝐴 ∈ V → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
6	1, 5	ax-mp 7	1 ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1285  Ⅎwnf 1390   ∈ 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:  elab2a  11026  bj-bdfindis  11184