Theorem rexanid 3251

Description: Cancellation law for restricted existential quantification. (Contributed by Peter Mazsa, 24-May-2018.) (Proof shortened by Wolf Lammen, 8-Jul-2023.)

Assertion

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

Proof of Theorem rexanid

Step	Hyp	Ref	Expression
1		ibar 531	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 ↔ (𝑥 ∈ 𝐴 ∧ 𝜑)))
2	1	bicomd 225	. 2 ⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ 𝜑))
3	2	rexbiia 3245	1 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 208   ∧ wa 398   ∈ wcel 2113  ∃wrex 3138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1780  df-rex 3143

This theorem is referenced by:  sn-axrep5v  39184