Theorem r19.29rOLD 3256

Description: Obsolete version of r19.29r 3255 as of 29-Jun-2023. (Contributed by NM, 31-Aug-1999.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
r19.29rOLD	⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Proof of Theorem r19.29rOLD

Step	Hyp	Ref	Expression
1		r19.29 3254	. 2 ⊢ ((∀𝑥 ∈ 𝐴 𝜓 ∧ ∃𝑥 ∈ 𝐴 𝜑) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑))
2		ancom 463	. 2 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜓 ∧ ∃𝑥 ∈ 𝐴 𝜑))
3		ancom 463	. . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜓 ∧ 𝜑))
4	3	rexbii 3247	. 2 ⊢ (∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜑))
5	1, 2, 4	3imtr4i 294	1 ⊢ ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) → ∃𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ∧ wa 398  ∀wral 3138  ∃wrex 3139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-ral 3143  df-rex 3144

This theorem is referenced by: (None)