Theorem clelsb4 2243

Description: Substitution applied to an atomic wff (class version of elsb4 1950). (Contributed by Jim Kingdon, 22-Nov-2018.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem clelsb4

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1508	. . 3 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑤
2	1	sbco2 1936	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝐴 ∈ 𝑤 ↔ [𝑦 / 𝑤]𝐴 ∈ 𝑤)
3		nfv 1508	. . . 4 ⊢ Ⅎ𝑤 𝐴 ∈ 𝑥
4		eleq2 2201	. . . 4 ⊢ (𝑤 = 𝑥 → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑥))
5	3, 4	sbie 1764	. . 3 ⊢ ([𝑥 / 𝑤]𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑥)
6	5	sbbii 1738	. 2 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝐴 ∈ 𝑤 ↔ [𝑦 / 𝑥]𝐴 ∈ 𝑥)
7		nfv 1508	. . 3 ⊢ Ⅎ𝑤 𝐴 ∈ 𝑦
8		eleq2 2201	. . 3 ⊢ (𝑤 = 𝑦 → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑦))
9	7, 8	sbie 1764	. 2 ⊢ ([𝑦 / 𝑤]𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝑦)
10	2, 6, 9	3bitr3i 209	1 ⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)

Syntax hints:   ↔ wb 104   ∈ wcel 1480  [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-cleq 2130  df-clel 2133

This theorem is referenced by:  peano1  4503  peano2  4504