Theorem elabf2 11111

Description: One implication of elabf 2750. (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 2225	. . 3 ⊢ Ⅎ𝑥𝐴
3		elabf2.nf	. . 3 ⊢ Ⅎ𝑥𝜓
4		elabf2.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))
5	2, 3, 4	elabgf2 11109	. 2 ⊢ (𝐴 ∈ V → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
6	1, 5	ax-mp 7	1 ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1287  Ⅎwnf 1392   ∈ wcel 1436  {cab 2071  Vcvv 2615

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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067

This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617

This theorem is referenced by:  elab2a  11113  bj-bdfindis  11271