Theorem elabf2 16483

Description: One implication of elabf 2950. (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 2375	. . 3 ⊢ Ⅎ𝑥𝐴
3		elabf2.nf	. . 3 ⊢ Ⅎ𝑥𝜓
4		elabf2.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))
5	2, 3, 4	elabgf2 16481	. 2 ⊢ (𝐴 ∈ V → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
6	1, 5	ax-mp 5	1 ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1398  Ⅎwnf 1509   ∈ wcel 2202  {cab 2217  Vcvv 2803

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805

This theorem is referenced by:  elab2a  16485  bj-bdfindis  16646