Theorem reximdd 44306

Description: Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

Hypotheses

Ref	Expression
reximdd.1	⊢ Ⅎ𝑥𝜑
reximdd.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒)
reximdd.3	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
reximdd	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)

Proof of Theorem reximdd

Step	Hyp	Ref	Expression
1		reximdd.3	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
2		reximdd.1	. . 3 ⊢ Ⅎ𝑥𝜑
3		reximdd.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜓) → 𝜒)
4	3	3exp 1118	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜓 → 𝜒)))
5	2, 4	reximdai 3257	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))
6	1, 5	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)

Syntax hints:   → wi 4   ∧ w3a 1086  Ⅎwnf 1784   ∈ wcel 2105  ∃wrex 3069

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 1912  ax-6 1970  ax-7 2010  ax-12 2170

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1088  df-ex 1781  df-nf 1785  df-ral 3061  df-rex 3070

This theorem is referenced by:  iinss2d  44316  xlimmnfvlem2  45011  xlimmnfv  45012  xlimpnfvlem2  45015  xlimpnfv  45016  fsupdm  46020  finfdm  46024