Theorem rexlimddv2 42111

Description: Restricted existential elimination rule of natural deduction. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

Hypotheses

Ref	Expression
rexlimddv2.1	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
rexlimddv2.2	⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)

Assertion

Ref	Expression
rexlimddv2	⊢ (𝜑 → 𝜒)

Distinct variable groups:   𝜒,𝑥   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝐴(𝑥)

Proof of Theorem rexlimddv2

Step	Hyp	Ref	Expression
1		rexlimddv2.1	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
2		rexlimddv2.2	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
3	2	anasss 469	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜒)
4	1, 3	rexlimddv 3293	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114  ∃wrex 3141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-ral 3145  df-rex 3146

This theorem is referenced by:  climxlim2lem  42133  xlimliminflimsup  42150