Theorem clelsb2 2312

Description: Substitution for the second argument of the membership predicate in an atomic formula (class version of elsb2 2185). (Contributed by Jim Kingdon, 22-Nov-2018.)

Assertion

Ref	Expression
clelsb2	⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem clelsb2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1552	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑤
2	1	sbco2 1994	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝐴 ∈ 𝑤 ↔ [𝑦 / 𝑤]𝐴 ∈ 𝑤)
3		nfv 1552	. . . 4 ⊢ Ⅎ𝑤 𝐴 ∈ 𝑥
4		eleq2 2270	. . . 4 ⊢ (𝑤 = 𝑥 → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑥))
5	3, 4	sbie 1815	. . 3 ⊢ ([𝑥 / 𝑤]𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑥)
6	5	sbbii 1789	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝐴 ∈ 𝑤 ↔ [𝑦 / 𝑥]𝐴 ∈ 𝑥)
7		nfv 1552	. . 3 ⊢ Ⅎ𝑤 𝐴 ∈ 𝑦
8		eleq2 2270	. . 3 ⊢ (𝑤 = 𝑦 → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑦))
9	7, 8	sbie 1815	. 2 ⊢ ([𝑦 / 𝑤]𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑦)
10	2, 6, 9	3bitr3i 210	1 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)

Syntax hints:   ↔ wb 105  [wsb 1786   ∈ wcel 2177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-cleq 2199  df-clel 2202

This theorem is referenced by:  peano1  4647  peano2  4648