Theorem elab3gf 3672

Description: Membership in a class abstraction, with a weaker antecedent than elabgf 3662. (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 3662	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
5	4	ibi 266	. . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)
6		pm2.21 123	. . 3 ⊢ (¬ 𝜓 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
7	5, 6	impbid2 225	. 2 ⊢ (¬ 𝜓 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
8	1, 2, 3	elabgf 3662	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
9	7, 8	ja 186	1 ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1534  Ⅎwnf 1778   ∈ wcel 2099  {cab 2703  Ⅎwnfc 2876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-v 3465

This theorem is referenced by: (None)