Theorem elsb4 2516

Description: Substitution applied to an atomic membership wff. For a version requiring more disjoint variables, but fewer axioms, see elsb4v 2515. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) Remove dependency on ax-11 2150. (Revised by Wolf Lammen, 28-Jul-2022.)

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		sbcom3 2487	. . 3 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑧 ∈ 𝑦 ↔ [𝑥 / 𝑤][𝑥 / 𝑦]𝑧 ∈ 𝑦)
2		elsb4v 2515	. . . 4 ⊢ ([𝑤 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑤)
3	2	sbbii 2019	. . 3 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑧 ∈ 𝑦 ↔ [𝑥 / 𝑤]𝑧 ∈ 𝑤)
4		nfv 1957	. . . 4 ⊢ Ⅎ𝑤[𝑥 / 𝑦]𝑧 ∈ 𝑦
5	4	sbf 2456	. . 3 ⊢ ([𝑥 / 𝑤][𝑥 / 𝑦]𝑧 ∈ 𝑦 ↔ [𝑥 / 𝑦]𝑧 ∈ 𝑦)
6	1, 3, 5	3bitr3i 293	. 2 ⊢ ([𝑥 / 𝑤]𝑧 ∈ 𝑤 ↔ [𝑥 / 𝑦]𝑧 ∈ 𝑦)
7		elsb4v 2515	. 2 ⊢ ([𝑥 / 𝑤]𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥)
8	6, 7	bitr3i 269	1 ⊢ ([𝑥 / 𝑦]𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑥)

Syntax hints:   ↔ wb 198  [wsb 2011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-12 2163  ax-13 2334

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ex 1824  df-nf 1828  df-sb 2012

This theorem is referenced by:  nfnid  5087