Theorem equsb3 2510

Description: Substitution in an equality. For a version requiring disjoint variables, but fewer axioms, see equsb3v 2291. (Contributed by Raph Levien and FL, 4-Dec-2005.) Remove dependency on ax-11 2150. (Revised by Wolf Lammen, 21-Sep-2018.)

Assertion

Ref	Expression
equsb3	⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧)

Distinct variable group:   𝑦,𝑧

Proof of Theorem equsb3

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcom3 2487	. . 3 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤][𝑥 / 𝑦]𝑦 = 𝑧)
2		equsb3v 2291	. . . 4 ⊢ ([𝑤 / 𝑦]𝑦 = 𝑧 ↔ 𝑤 = 𝑧)
3	2	sbbii 2019	. . 3 ⊢ ([𝑥 / 𝑤][𝑤 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑤]𝑤 = 𝑧)
4		nfv 1957	. . . 4 ⊢ Ⅎ𝑤[𝑥 / 𝑦]𝑦 = 𝑧
5	4	sbf 2456	. . 3 ⊢ ([𝑥 / 𝑤][𝑥 / 𝑦]𝑦 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
6	1, 3, 5	3bitr3i 293	. 2 ⊢ ([𝑥 / 𝑤]𝑤 = 𝑧 ↔ [𝑥 / 𝑦]𝑦 = 𝑧)
7		equsb3v 2291	. 2 ⊢ ([𝑥 / 𝑤]𝑤 = 𝑧 ↔ 𝑥 = 𝑧)
8	6, 7	bitr3i 269	1 ⊢ ([𝑥 / 𝑦]𝑦 = 𝑧 ↔ 𝑥 = 𝑧)

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-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:  mo3OLD  2581  sb8iota  6108  mo5f  29913  mptsnunlem  33788  wl-equsb3  33939  wl-mo3t  33959  wl-sb8eut  33960  frege55lem1b  39159  sbeqal1  39568