Theorem bj-equsvt 34647

Description: A variant of equsv 2012. (Contributed by BJ, 7-Oct-2024.)

Assertion

Ref	Expression
bj-equsvt	⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-equsvt

Step	Hyp	Ref	Expression
1		bj-19.23t 34638	. 2 ⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ (∃𝑥 𝑥 = 𝑦 → 𝜑)))
2		ax6ev 1978	. . 3 ⊢ ∃𝑥 𝑥 = 𝑦
3	2	a1bi 366	. 2 ⊢ (𝜑 ↔ (∃𝑥 𝑥 = 𝑦 → 𝜑))
4	1, 3	bitr4di 292	1 ⊢ (Ⅎ'𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541  ∃wex 1787  Ⅎ'wnnf 34591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-6 1976

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-bj-nnf 34592

This theorem is referenced by:  bj-equsalvwd  34648