Theorem elabgf2 10590

Description: One implication of elabgf 2736. (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 2221	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
4	1, 3	nfel 2227	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜑}
5	2, 4	nfim 1504	. 2 ⊢ Ⅎ𝑥(𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})
6		elabgf0 10587	. 2 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))
7		bicom1 129	. . 3 ⊢ ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))
8		elabgf2.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))
9		bi1 116	. . . 4 ⊢ ((𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}) → (𝜑 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
10	8, 9	syl9 71	. . 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 10586	1 ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))

Syntax hints:   → wi 4   ↔ wb 103   = wceq 1284  Ⅎwnf 1389   ∈ wcel 1433  {cab 2067  Ⅎwnfc 2206

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by:  elabf2  10592  elabg2  10595  bj-intabssel1  10600