Theorem elab3g 3669

Description: Membership in a class abstraction, with a weaker antecedent than elabg 3660. (Contributed by NM, 29-Aug-2006.)

Hypothesis

Ref	Expression
elab3g.1	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
elab3g	⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))

Distinct variable groups:   𝜓,𝑥   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem elab3g

Step	Hyp	Ref	Expression
1		elab3g.1	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
2	1	elabg 3660	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
3	2	ibi 267	. . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)
4		pm2.21 123	. . 3 ⊢ (¬ 𝜓 → (𝜓 → 𝐴 ∈ {𝑥 ∣ 𝜑}))
5	3, 4	impbid2 226	. 2 ⊢ (¬ 𝜓 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
6	1	elabg 3660	. 2 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
7	5, 6	ja 186	1 ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {cab 2714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810

This theorem is referenced by:  elab3  3670  elssabg  5318  elrnmptg  5946  elrelimasn  6078  elmapg  8858  isust  24147  ellimc  25831  isismty  37830  clublem  43601