Theorem elab3gf 3356

Description: Membership in a class abstraction, with a weaker antecedent than elabgf 3348. (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	. . . . 5 ⊢ Ⅎ𝑥𝐴
2		elab3gf.2	. . . . 5 ⊢ Ⅎ𝑥𝜓
3		elab3gf.3	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	1, 2, 3	elabgf 3348	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
5	4	ibi 256	. . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)
6		pm2.21 120	. . 3 ⊢ (¬ 𝜓 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
7	5, 6	impbid2 216	. 2 ⊢ (¬ 𝜓 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
8	1, 2, 3	elabgf 3348	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
9	7, 8	ja 173	1 ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   = wceq 1483  Ⅎwnf 1708   ∈ wcel 1990  {cab 2608  Ⅎwnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  elab3g  3357