Theorem elabgf2 13815

Description: One implication of elabgf 2872. (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 2314	. . . 4 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
4	1, 3	nfel 2321	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜑}
5	2, 4	nfim 1565	. 2 ⊢ Ⅎ𝑥(𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑})
6		elabgf0 13812	. 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 13811	1 ⊢ (𝐴 ∈ 𝐵 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1348  Ⅎwnf 1453   ∈ wcel 2141  {cab 2156  Ⅎwnfc 2299

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:  elabf2  13817  elabg2  13820  bj-intabssel1  13825