Theorem elabf2 15428

Description: One implication of elabf 2907. (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 2339	. . 3 ⊢ Ⅎ𝑥𝐴
3		elabf2.nf	. . 3 ⊢ Ⅎ𝑥𝜓
4		elabf2.1	. . 3 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))
5	2, 3, 4	elabgf2 15426	. 2 ⊢ (𝐴 ∈ V → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
6	1, 5	ax-mp 5	1 ⊢ (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})

Syntax hints:   → wi 4   = wceq 1364  Ⅎwnf 1474   ∈ wcel 2167  {cab 2182  Vcvv 2763

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  elab2a  15430  bj-bdfindis  15593