Theorem wl-lem-exsb 37568

Description: This theorem provides a basic working step in proving theorems about ∃* or ∃!. (Contributed by Wolf Lammen, 3-Oct-2019.)

Assertion

Ref	Expression
wl-lem-exsb	⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem wl-lem-exsb

Step	Hyp	Ref	Expression
1		ax12v2 2178	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		sp 2182	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → (𝑥 = 𝑦 → 𝜑))
3	2	com12 32	. 2 ⊢ (𝑥 = 𝑦 → (∀𝑥(𝑥 = 𝑦 → 𝜑) → 𝜑))
4	1, 3	impbid 212	1 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-12 2176

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779

This theorem is referenced by: (None)