Theorem elabf2 13817

Description: One implication of elabf 2873. (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 2312	. . 3 ⊢ Ⅎ𝑥𝐴
3		elabf2.nf	. . 3 ⊢ Ⅎ𝑥𝜓
4		elabf2.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))
5	2, 3, 4	elabgf2 13815	. 2 ⊢ (𝐴 ∈ V → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
6	1, 5	ax-mp 5	1 ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1348  Ⅎwnf 1453   ∈ wcel 2141  {cab 2156  Vcvv 2730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  elab2a  13819  bj-bdfindis  13982