Theorem elsb2 2184

Description: Substitution for the second argument of the non-logical predicate in an atomic formula. See elsb1 2183 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 1549	. . . . 5 ⊢ (𝑧 ∈ 𝑥 → ∀𝑤 𝑧 ∈ 𝑥)
2		elequ2 2181	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥))
3	1, 2	sbieh 1813	. . . 4 ⊢ ([𝑥 / 𝑤]𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥)
4	3	sbbii 1788	. . 3 ⊢ ([𝑦 / 𝑥][𝑥 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑦 / 𝑥]𝑧 ∈ 𝑥)
5		ax-17 1549	. . . 4 ⊢ (𝑧 ∈ 𝑤 → ∀𝑥 𝑧 ∈ 𝑤)
6	5	sbco2h 1992	. . 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 1808	. . . 4 ⊢ [𝑦 / 𝑤]𝑤 = 𝑦
9		elequ2 2181	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦))
10	9	sbimi 1787	. . . 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 1987	. . 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 1549	. . 3 ⊢ (𝑧 ∈ 𝑦 → ∀𝑤 𝑧 ∈ 𝑦)
15	14	sbh 1799	. 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 1785

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786

This theorem is referenced by: (None)