Theorem elabgf2 15649

Description: One implication of elabgf 2914. (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 2349	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
4	1, 3	nfel 2356	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜑}
5	2, 4	nfim 1594	. 2 ⊢ Ⅎ𝑥(𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})
6		elabgf0 15646	. 2 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))
7		bicom1 131	. . 3 ⊢ ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))
8		elabgf2.1	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 → 𝜑))
9		biimp 118	. . . 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 15645	1 ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1372  Ⅎwnf 1482   ∈ wcel 2175  {cab 2190  Ⅎwnfc 2334

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773

This theorem is referenced by:  elabf2  15651  elabg2  15654  bj-intabssel1  15659