Theorem clelsb3fOLD 2937

Description: Obsolete version of clelsb3f 2936 as of 7-May-2023. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
clelsb3f.1	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
clelsb3fOLD	⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)

Proof of Theorem clelsb3fOLD

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clelsb3f.1	. . . 4 ⊢ Ⅎ𝑦𝐴
2	1	nfcri 2926	. . 3 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
3	2	sbco2 2477	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑤]𝑤 ∈ 𝐴)
4		nfv 1873	. . . 4 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐴
5		eleq1w 2848	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
6	4, 5	sbie 2468	. . 3 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
7	6	sbbii 2027	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑥 / 𝑦]𝑦 ∈ 𝐴)
8		nfv 1873	. . 3 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
9		eleq1w 2848	. . 3 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
10	8, 9	sbie 2468	. 2 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
11	3, 7, 10	3bitr3i 293	1 ⊢ ([𝑥 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)

Syntax hints:   ↔ wb 198  [wsb 2015   ∈ wcel 2050  Ⅎwnfc 2916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clel 2846  df-nfc 2918

This theorem is referenced by: (None)