Theorem clelsb3fOLD 2986

Description: Obsolete version of clelsb3f 2985 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	⊢ Ⅎ𝑥𝐴

Assertion

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

Proof of Theorem clelsb3fOLD

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clelsb3f.1	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	nfcri 2974	. . 3 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴
3	2	sbco2 2552	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑤]𝑤 ∈ 𝐴)
4		nfv 1914	. . . 4 ⊢ Ⅎ𝑤 𝑥 ∈ 𝐴
5		eleq1w 2898	. . . 4 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
6	4, 5	sbie 2543	. . 3 ⊢ ([𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)
7	6	sbbii 2080	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑤 ∈ 𝐴 ↔ [𝑦 / 𝑥]𝑥 ∈ 𝐴)
8		nfv 1914	. . 3 ⊢ Ⅎ𝑤 𝑦 ∈ 𝐴
9		eleq1w 2898	. . 3 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
10	8, 9	sbie 2543	. 2 ⊢ ([𝑦 / 𝑤]𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
11	3, 7, 10	3bitr3i 303	1 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)

Syntax hints:   ↔ wb 208  [wsb 2068   ∈ wcel 2113  Ⅎwnfc 2964

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-10 2144  ax-11 2160  ax-12 2176  ax-13 2389

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clel 2896  df-nfc 2966

This theorem is referenced by: (None)