Theorem reximddv3 3182

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

Hypotheses

Ref	Expression
reximddv3.1	⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
reximddv3.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
reximddv3	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem reximddv3

Step	Hyp	Ref	Expression
1		reximddv3.1	. . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝜓) → 𝜒)
2	1	anasss 471	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜒)
3		reximddv3.2	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
4	2, 3	reximddv 3181	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2145  ∃wrex 3089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-rex 3090

This theorem is referenced by:  rprmasso2  33733  rprmirredlem  33737  rnmptlb  45816  rnmptbddlem  45817  limclner  46223  climisp  46318  climrescn  46320  liminflbuz2  46387  liminflimsupxrre  46389  climxlim2lem  46417  hoicvr  47120