Theorem eubi 2585

Description: Equivalence theorem for the unique existential quantifier. Theorem *14.271 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) Reduce dependencies on axioms. (Revised by BJ, 7-Oct-2022.)

Assertion

Ref	Expression
eubi	⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓))

Proof of Theorem eubi

Step	Hyp	Ref	Expression
1		exbi 1849	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
2		mobi 2548	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃*𝑥𝜑 ↔ ∃*𝑥𝜓))
3	1, 2	anbi12d 633	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → ((∃𝑥𝜑 ∧ ∃*𝑥𝜑) ↔ (∃𝑥𝜓 ∧ ∃*𝑥𝜓)))
4		df-eu 2570	. 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑))
5		df-eu 2570	. 2 ⊢ (∃!𝑥𝜓 ↔ (∃𝑥𝜓 ∧ ∃*𝑥𝜓))
6	3, 4, 5	3bitr4g 314	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃!𝑥𝜑 ↔ ∃!𝑥𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540  ∃wex 1781  ∃*wmo 2538  ∃!weu 2569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-mo 2540  df-eu 2570

This theorem is referenced by:  eubii  2586  eubidv  2587  eubid  2588