Theorem rexanidOLD 3252

Description: Obsolete version of rexanid 3251 as of 8-Jul-2023. (Contributed by Peter Mazsa, 24-May-2018.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem rexanidOLD

Step	Hyp	Ref	Expression
1		anabs5 661	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	exbii 1847	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-rex 3143	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
4		df-rex 3143	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5	2, 3, 4	3bitr4i 305	1 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 208   ∧ wa 398  ∃wex 1779   ∈ 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: (None)