Theorem bj-equsal1t 36788

Description: Duplication of wl-equsal1t 37496, with shorter proof. If one imposes a disjoint variable condition on x,y , then one can use alequexv 2000 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 37497 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 36750	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
2		19.9t 2205	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	imbitrid 244	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
4		nf5r 2195	. . 3 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
5		ala1 1811	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
6	4, 5	syl6 35	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
7	3, 6	impbid 212	1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535  ∃wex 1777  Ⅎwnf 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-12 2178  ax-13 2380

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-nf 1782

This theorem is referenced by:  bj-equsal1ti  36789