Theorem rmoanid 3379

Description: Cancellation law for restricted at-most-one quantification. (Contributed by Peter Mazsa, 24-May-2018.) (Proof shortened by Wolf Lammen, 12-Jan-2025.)

Assertion

Ref	Expression
rmoanid	⊢ (∃*𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥 ∈ 𝐴 𝜑)

Proof of Theorem rmoanid

Step	Hyp	Ref	Expression
1		ibar 537	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
2	1	bicomd 226	. 2 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝜑))
3	2	rmobiia 3375	1 ⊢ (∃*𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2143  ∃*wrmo 3368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-mo 2567  df-rmo 3369

This theorem is referenced by:  raldmqseu  39014