Theorem elab2gw 40416

Description: Membership in a class abstraction, using two substitution hypotheses to avoid a disjoint variable condition on 𝑥 and 𝐴, which is not usually significant since 𝐵 is usually a constant. (Contributed by SN, 16-May-2024.)

Hypotheses

Ref	Expression
elabgw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
elabgw.2	⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))
elab2gw.3	⊢ 𝐵 = {𝑥 ∣ 𝜑}

Assertion

Ref	Expression
elab2gw	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ 𝜒))

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦   𝜑,𝑦   𝜒,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝜒(𝑥)   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem elab2gw

Step	Hyp	Ref	Expression
1		elab2gw.3	. . 3 ⊢ 𝐵 = {𝑥 ∣ 𝜑}
2	1	eleq2i 2828	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
3		elabgw.1	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4		elabgw.2	. . 3 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))
5	3, 4	elabgw 40415	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒))
6	2, 5	bitrid 282	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1540   ∈ wcel 2105  {cab 2713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814

This theorem is referenced by:  elrab2w  40417