Theorem elab3gf 2914

Description: Membership in a class abstraction, with a weaker antecedent than elabgf 2906. (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 2906	. . 3 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
5	4	ibi 176	. 2 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)
6	1, 2, 3	elabgf 2906	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))
7	6	imim2i 12	. . 3 ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝜓 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)))
8		biimpr 130	. . 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 143	1 ⊢ ((𝜓 → 𝐴 ∈ 𝐵) → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364  Ⅎwnf 1474   ∈ wcel 2167  {cab 2182  Ⅎwnfc 2326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  elab3g  2915