Theorem elsb4 2136

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	⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)

Distinct variable group:   𝑥,𝑧

Proof of Theorem elsb4

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1506	. . . . 5 ⊢ (𝑧 ∈ 𝑥 → ∀𝑤 𝑧 ∈ 𝑥)
2		elequ2 2133	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥))
3	1, 2	sbieh 1770	. . . 4 ⊢ ([𝑥 / 𝑤]𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥)
4	3	sbbii 1745	. . 3 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝑥)
5		ax-17 1506	. . . 4 ⊢ (𝑧 ∈ 𝑤 → ∀𝑥 𝑧 ∈ 𝑤)
6	5	sbco2h 1944	. . 3 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑦 / 𝑤]𝑧 ∈ 𝑤)
7	4, 6	bitr3i 185	. 2 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ [𝑦 / 𝑤]𝑧 ∈ 𝑤)
8		equsb1 1765	. . . 4 ⊢ [𝑦 / 𝑤]𝑤 = 𝑦
9		elequ2 2133	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦))
10	9	sbimi 1744	. . . 4 ⊢ ([𝑦 / 𝑤]𝑤 = 𝑦 → [𝑦 / 𝑤](𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦))
11	8, 10	ax-mp 5	. . 3 ⊢ [𝑦 / 𝑤](𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦)
12		sbbi 1939	. . 3 ⊢ ([𝑦 / 𝑤](𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) ↔ ([𝑦 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑦 / 𝑤]𝑧 ∈ 𝑦))
13	11, 12	mpbi 144	. 2 ⊢ ([𝑦 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑦 / 𝑤]𝑧 ∈ 𝑦)
14		ax-17 1506	. . 3 ⊢ (𝑧 ∈ 𝑦 → ∀𝑤 𝑧 ∈ 𝑦)
15	14	sbh 1756	. 2 ⊢ ([𝑦 / 𝑤]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦)
16	7, 13, 15	3bitri 205	1 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)

Syntax hints:   ↔ wb 104  [wsb 1742

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743

This theorem is referenced by: (None)