Theorem equsb3r 2140

Description: Substitution applied to the atomic wff with equality. Variant of equsb3 2139. (Contributed by AV, 29-Jul-2023.) (Proof shortened by Wolf Lammen, 2-Sep-2023.)

Assertion

Ref	Expression
equsb3r	⊢ ([𝑦 / 𝑥]𝑧 = 𝑥 ↔ 𝑧 = 𝑦)

Distinct variable group:   𝑥,𝑧

Proof of Theorem equsb3r

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equequ2 2058	. 2 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑤))
2		equequ2 2058	. 2 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦))
3	1, 2	sbievw2 2134	1 ⊢ ([𝑦 / 𝑥]𝑧 = 𝑥 ↔ 𝑧 = 𝑦)

This proof depends on syntax axioms:   ↔ wb 209  [wsb 2098

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099

This theorem is used by:  icheq  48638