Theorem clelsb2OLD 2869

Description: Obsolete version of clelsb2 2868 as of 24-Nov-2024.) (Contributed by Jim Kingdon, 22-Nov-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
clelsb2OLD	⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem clelsb2OLD

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1913	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑤
2	1	sbco2 2515	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝐴 ∈ 𝑤 ↔ [𝑦 / 𝑤]𝐴 ∈ 𝑤)
3		nfv 1913	. . . 4 ⊢ Ⅎ𝑤 𝐴 ∈ 𝑥
4		eleq2 2829	. . . 4 ⊢ (𝑤 = 𝑥 → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑥))
5	3, 4	sbie 2506	. . 3 ⊢ ([𝑥 / 𝑤]𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑥)
6	5	sbbii 2075	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝐴 ∈ 𝑤 ↔ [𝑦 / 𝑥]𝐴 ∈ 𝑥)
7		nfv 1913	. . 3 ⊢ Ⅎ𝑤 𝐴 ∈ 𝑦
8		eleq2 2829	. . 3 ⊢ (𝑤 = 𝑦 → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑦))
9	7, 8	sbie 2506	. 2 ⊢ ([𝑦 / 𝑤]𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑦)
10	2, 6, 9	3bitr3i 301	1 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)

Syntax hints:   ↔ wb 206  [wsb 2063   ∈ wcel 2107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2376  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-cleq 2728  df-clel 2815

This theorem is referenced by: (None)