Theorem equsv 2009

Description: If a formula does not contain a variable 𝑥, then it is equivalent to the corresponding prototype of substitution with a fresh variable (see sb6 2093). (Contributed by BJ, 23-Jul-2023.)

Assertion

Ref	Expression
equsv	⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑥

Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem equsv

Step	Hyp	Ref	Expression
1		19.23v 1943	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ (∃𝑥 𝑥 = 𝑦 → 𝜑))
2		ax6ev 1972	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
3	2	a1bi 365	. 2 ⊢ (𝜑 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜑))
4	1, 3	bitr4i 280	1 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535  ∃wex 1780

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

This theorem depends on definitions:  df-bi 209  df-ex 1781

This theorem is referenced by:  equsalvw  2010