Theorem sn-elabg 39187

Description: Membership in a class abstraction, using implicit substitution and an intermediate setvar 𝑦 to avoid ax-10 2144, ax-11 2160, ax-12 2176. It also avoids a disjoint variable condition on 𝑥 and 𝐴. Compare sbievw2 2106. (Contributed by SN, 20-Apr-2024.)

Hypotheses

Ref	Expression
sn-elabg.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
sn-elabg.2	⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
sn-elabg	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 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)   𝑉(𝑥,𝑦)

Proof of Theorem sn-elabg

Step	Hyp	Ref	Expression
1		eleq1 2899	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝐴 ∈ {𝑥 ∣ 𝜑}))
2		sn-elabg.2	. . 3 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒))
3	1, 2	bibi12d 348	. 2 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓) ↔ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒)))
4		df-clab 2799	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑)
5		sn-elabg.1	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
6	5	sbievw 2102	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
7	4, 6	bitri 277	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ 𝜓)
8	3, 7	vtoclg 3564	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1536  [wsb 2068   ∈ wcel 2113  {cab 2798

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892

This theorem is referenced by: (None)