Theorem rmoanidOLD 3391

Description: Obsolete version of rmoanid 3389 as of 12-Jan-2025. (Contributed by Peter Mazsa, 24-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem rmoanidOLD

Step	Hyp	Ref	Expression
1		anabs5 663	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ 𝜑))
2	1	mobii 2547	. 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
3		df-rmo 3379	. 2 ⊢ (∃*𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
4		df-rmo 3379	. 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))
5	2, 3, 4	3bitr4i 303	1 ⊢ (∃*𝑥 ∈ 𝐴 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃*𝑥 ∈ 𝐴 𝜑)

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2107  ∃*wmo 2537  ∃*wrmo 3378

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

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-mo 2539  df-rmo 3379

This theorem is referenced by: (None)