Theorem bj-equsal1t 33388

Description: Duplication of wl-equsal1t 33924, with shorter proof. If one imposes a disjoint variable condition on x,y , then one can use bj-alequexv 33248 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 33925 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 33300	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
2		19.9t 2189	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	syl5ib 236	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
4		nf5r 2178	. . 3 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
5		ala1 1857	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
6	4, 5	syl6 35	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
7	3, 6	impbid 204	1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1599  ∃wex 1823  Ⅎwnf 1827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-12 2163  ax-13 2334

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-nf 1828

This theorem is referenced by:  bj-equsal1ti  33389