Theorem elsb4 1853

Description: Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
elsb4	⊢ ([𝑥 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥)

Distinct variable group:   𝑦,𝑧

Proof of Theorem elsb4

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1419	. . . . 5 ⊢ (𝑧 ∈ 𝑦 → ∀𝑤 𝑧 ∈ 𝑦)
2		elequ2 1601	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦))
3	1, 2	sbieh 1673	. . . 4 ⊢ ([𝑦 / 𝑤]𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦)
4	3	sbbii 1648	. . 3 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑥 / 𝑦]𝑧 ∈ 𝑦)
5		ax-17 1419	. . . 4 ⊢ (𝑧 ∈ 𝑤 → ∀𝑦 𝑧 ∈ 𝑤)
6	5	sbco2h 1838	. . 3 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑥 / 𝑤]𝑧 ∈ 𝑤)
7	4, 6	bitr3i 175	. 2 ⊢ ([𝑥 / 𝑦]𝑧 ∈ 𝑦 ↔ [𝑥 / 𝑤]𝑧 ∈ 𝑤)
8		equsb1 1668	. . . 4 ⊢ [𝑥 / 𝑤]𝑤 = 𝑥
9		elequ2 1601	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥))
10	9	sbimi 1647	. . . 4 ⊢ ([𝑥 / 𝑤]𝑤 = 𝑥 → [𝑥 / 𝑤](𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥))
11	8, 10	ax-mp 7	. . 3 ⊢ [𝑥 / 𝑤](𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥)
12		sbbi 1833	. . 3 ⊢ ([𝑥 / 𝑤](𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥) ↔ ([𝑥 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑥 / 𝑤]𝑧 ∈ 𝑥))
13	11, 12	mpbi 133	. 2 ⊢ ([𝑥 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑥 / 𝑤]𝑧 ∈ 𝑥)
14		ax-17 1419	. . 3 ⊢ (𝑧 ∈ 𝑥 → ∀𝑤 𝑧 ∈ 𝑥)
15	14	sbh 1659	. 2 ⊢ ([𝑥 / 𝑤]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥)
16	7, 13, 15	3bitri 195	1 ⊢ ([𝑥 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥)

Syntax hints:   ↔ wb 98  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by:  peano2  4318