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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 2090). (Contributed by BJ, 23-Jul-2023.)
Assertion
Ref Expression
equsv (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑)
Distinct variable groups:   𝑥,𝑦   𝜑,𝑥
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem equsv
StepHypRef Expression
1 19.23v 1943 . 2 (∀𝑥(𝑥 = 𝑦𝜑) ↔ (∃𝑥 𝑥 = 𝑦𝜑))
2 ax6ev 1972 . . 3 𝑥 𝑥 = 𝑦
32a1bi 366 . 2 (𝜑 ↔ (∃𝑥 𝑥 = 𝑦𝜑))
41, 3bitr4i 281 1 (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  ∃wex 1781 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970 This theorem depends on definitions:  df-bi 210  df-ex 1782 This theorem is referenced by:  equsalvw  2010
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