Theorem clelsb4 2194

Description: Substitution applied to an atomic wff (class version of elsb4 1902). (Contributed by Jim Kingdon, 22-Nov-2018.)

Assertion

Ref	Expression
clelsb4	⊢ ([𝑥 / 𝑦]𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥)

Distinct variable group:   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem clelsb4

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1467	. . 3 ⊢ Ⅎ𝑦 𝐴 ∈ 𝑤
2	1	sbco2 1888	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝐴 ∈ 𝑤 ↔ [𝑥 / 𝑤]𝐴 ∈ 𝑤)
3		nfv 1467	. . . 4 ⊢ Ⅎ𝑤 𝐴 ∈ 𝑦
4		eleq2 2152	. . . 4 ⊢ (𝑤 = 𝑦 → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑦))
5	3, 4	sbie 1722	. . 3 ⊢ ([𝑦 / 𝑤]𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑦)
6	5	sbbii 1696	. 2 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝐴 ∈ 𝑤 ↔ [𝑥 / 𝑦]𝐴 ∈ 𝑦)
7		nfv 1467	. . 3 ⊢ Ⅎ𝑤 𝐴 ∈ 𝑥
8		eleq2 2152	. . 3 ⊢ (𝑤 = 𝑥 → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑥))
9	7, 8	sbie 1722	. 2 ⊢ ([𝑥 / 𝑤]𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑥)
10	2, 6, 9	3bitr3i 209	1 ⊢ ([𝑥 / 𝑦]𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥)

Syntax hints:   ↔ wb 104   ∈ wcel 1439  [wsb 1693

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-cleq 2082  df-clel 2085

This theorem is referenced by:  peano1  4424  peano2  4425