Theorem elab3gf 2991

Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2984. (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 2984	. . . 4 ⊢ (A ∈ {x ∣ φ} → (A ∈ {x ∣ φ} ↔ ψ))
5	4	ibi 232	. . 3 ⊢ (A ∈ {x ∣ φ} → ψ)
6		pm2.21 100	. . 3 ⊢ (¬ ψ → (ψ → A ∈ {x ∣ φ}))
7	5, 6	impbid2 195	. 2 ⊢ (¬ ψ → (A ∈ {x ∣ φ} ↔ ψ))
8	1, 2, 3	elabgf 2984	. 2 ⊢ (A ∈ B → (A ∈ {x ∣ φ} ↔ ψ))
9	7, 8	ja 153	1 ⊢ ((ψ → A ∈ B) → (A ∈ {x ∣ φ} ↔ ψ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is used by:  elab3g  2992