Theorem elsb2 2156

Description: Substitution for the second argument of the non-logical predicate in an atomic formula. See elsb1 2155 for substitution for the first argument. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
elsb2	⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)

Distinct variable group:   𝑥,𝑧

Proof of Theorem elsb2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-17 1526	. . . . 5 ⊢ (𝑧 ∈ 𝑥 → ∀𝑤 𝑧 ∈ 𝑥)
2		elequ2 2153	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥))
3	1, 2	sbieh 1790	. . . 4 ⊢ ([𝑥 / 𝑤]𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥)
4	3	sbbii 1765	. . 3 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝑥)
5		ax-17 1526	. . . 4 ⊢ (𝑧 ∈ 𝑤 → ∀𝑥 𝑧 ∈ 𝑤)
6	5	sbco2h 1964	. . 3 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑦 / 𝑤]𝑧 ∈ 𝑤)
7	4, 6	bitr3i 186	. 2 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ [𝑦 / 𝑤]𝑧 ∈ 𝑤)
8		equsb1 1785	. . . 4 ⊢ [𝑦 / 𝑤]𝑤 = 𝑦
9		elequ2 2153	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦))
10	9	sbimi 1764	. . . 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 1959	. . 3 ⊢ ([𝑦 / 𝑤](𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) ↔ ([𝑦 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑦 / 𝑤]𝑧 ∈ 𝑦))
13	11, 12	mpbi 145	. 2 ⊢ ([𝑦 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑦 / 𝑤]𝑧 ∈ 𝑦)
14		ax-17 1526	. . 3 ⊢ (𝑧 ∈ 𝑦 → ∀𝑤 𝑧 ∈ 𝑦)
15	14	sbh 1776	. 2 ⊢ ([𝑦 / 𝑤]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑦)
16	7, 13, 15	3bitri 206	1 ⊢ ([𝑦 / 𝑥]𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)

Syntax hints:   ↔ wb 105  [wsb 1762

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763

This theorem is referenced by: (None)