Theorem equsb3r 2110

Description: Substitution applied to the atomic wff with equality. Variant of equsb3 2109. (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 2033	. 2 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝑥 ↔ 𝑧 = 𝑤))
2		equequ2 2033	. 2 ⊢ (𝑤 = 𝑦 → (𝑧 = 𝑤 ↔ 𝑧 = 𝑦))
3	1, 2	sbievw2 2107	1 ⊢ ([𝑦 / 𝑥]𝑧 = 𝑥 ↔ 𝑧 = 𝑦)

Syntax hints:   ↔ wb 208  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-sb 2070

This theorem is referenced by:  icheq  43669