Theorem elab3gf 2829

Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2821. (Contributed by NM, 6-Sep-2011.)

Hypotheses

Ref	Expression
elab3gf.1	⊢ Ⅎ𝑥𝐴
elab3gf.2	⊢ Ⅎ𝑥𝜓
elab3gf.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
elab3gf	⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))

Proof of Theorem elab3gf

Step	Hyp	Ref	Expression
1		elab3gf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2		elab3gf.2	. . . 4 ⊢ Ⅎ𝑥𝜓
3		elab3gf.3	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	elabgf 2821	. . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
5	4	ibi 175	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)
6	1, 2, 3	elabgf 2821	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
7	6	imim2i 12	. . 3 ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝜓 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)))
8		bi2 129	. . 3 ⊢ ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
9	7, 8	syli 37	. 2 ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
10	5, 9	impbid2 142	1 ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331  Ⅎwnf 1436   ∈ wcel 1480  {cab 2123  Ⅎwnfc 2266

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683

This theorem is referenced by:  elab3g  2830