Theorem elab2gw 3613

Description: Membership in a class abstraction, using implicit substitution and an intermediate setvar 𝑦 to avoid ax-10 2142, ax-11 2158, ax-12 2175. It also avoids a disjoint variable condition on 𝑥 and 𝐴. (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 2881	. 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 3612	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒))
6	2, 5	syl5bb 286	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {cab 2776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870

This theorem is referenced by:  eldif  3891  elin  3897  elun  4076  elpwg  4500  elsng  4539  elprg  4546  eluni  4803  elint  4844  elintg  4846  elxpi  5541  elong  6167  isfin2  9705  iswun  10115  elnpi  10399  issal  42956