Theorem elabgf2 13661

Description: One implication of elabgf 2868. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
elabgf2.nf1	⊢ Ⅎ𝑥𝐴
elabgf2.nf2	⊢ Ⅎ𝑥𝜓
elabgf2.1	⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))

Assertion

Ref	Expression
elabgf2	⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))

Proof of Theorem elabgf2

Step	Hyp	Ref	Expression
1		elabgf2.nf1	. 2 ⊢ Ⅎ𝑥𝐴
2		elabgf2.nf2	. . 3 ⊢ Ⅎ𝑥𝜓
3		nfab1 2310	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
4	1, 3	nfel 2317	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜑}
5	2, 4	nfim 1560	. 2 ⊢ Ⅎ𝑥(𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})
6		elabgf0 13658	. 2 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))
7		bicom1 130	. . 3 ⊢ ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))
8		elabgf2.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))
9		biimp 117	. . . 4 ⊢ ((𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) → (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
10	8, 9	syl9 72	. . 3 ⊢ (𝑥 = 𝐴 → ((𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})))
11	7, 10	syl5 32	. 2 ⊢ (𝑥 = 𝐴 → ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})))
12	1, 5, 6, 11	bj-vtoclgf 13657	1 ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1343  Ⅎwnf 1448   ∈ wcel 2136  {cab 2151  Ⅎwnfc 2295

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728

This theorem is referenced by:  elabf2  13663  elabg2  13666  bj-intabssel1  13671