Theorem bj-equsal1t 37552

Description: Duplication of wl-equsal1t 38292, with shorter proof. If one imposes a disjoint variable condition on 𝑥, 𝑦, then one can use alequexv 2034 and reduce axiom dependencies, and similarly for the following theorems. Note: wl-equsalcom 38293 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 37514	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∃𝑥𝜑)
2		19.9t 2242	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	imbitrid 247	. 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
4		nf5r 2232	. . 3 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
5		ala1 1846	. . 3 ⊢ (∀𝑥𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
6	4, 5	syl6 36	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
7	3, 6	impbid 215	1 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2215  ax-13 2403

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-nf 1817

This theorem is used by:  bj-equsal1ti  37553