Theorem equsb1v 2143

Description: Substitution applied to an atomic wff. Version of equsb1 2526 with a disjoint variable condition, which neither requires ax-12 2216 nor ax-13 2407. (Contributed by NM, 10-May-1993.) (Revised by BJ, 11-Sep-2019.) Remove dependencies on axioms. (Revised by Wolf Lammen, 30-May-2023.) (Proof shortened by Steven Nguyen, 19-Jun-2023.) Revise df-sb 2100. (Revised by Steven Nguyen, 11-Jul-2023.) (Proof shortened by Steven Nguyen, 22-Jul-2023.)

Assertion

Ref	Expression
equsb1v	⊢ [𝑦 / 𝑥]𝑥 = 𝑦

Distinct variable group:   𝑥,𝑦

Proof of Theorem equsb1v

Step	Hyp	Ref	Expression
1		equid 2045	. 2 ⊢ 𝑦 = 𝑦
2		equsb3 2141	. 2 ⊢ ([𝑦 / 𝑥]𝑥 = 𝑦 ↔ 𝑦 = 𝑦)
3	1, 2	mpbir 234	1 ⊢ [𝑦 / 𝑥]𝑥 = 𝑦

This proof depends on syntax axioms:  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100

This theorem is used by:  pm13.183  3628  exss  5449