Theorem bj-equsal1t 31997

Description: Duplication of wl-equsal1t 32506, with shorter proof. Note: wl-equsalcom 32507 is also interesting. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-equsal1t	⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))

Proof of Theorem bj-equsal1t

Step	Hyp	Ref	Expression
1		bj-alequex 31895	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
2		19.9t 2059	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	syl5ib 233	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
4		nf5r 2052	. . 3 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
5		ala1 1755	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
6	4, 5	syl6 34	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
7	3, 6	impbid 201	1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695  Ⅎwnf 1699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701

This theorem is referenced by:  bj-equsal1ti  31998