Theorem reximddv3 3179

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 472	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) → 𝜒)
3		reximddv3.2	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)
4	2, 3	reximddv 3178	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144  ∃wrex 3086

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-rex 3087

This theorem is used by:  inagcgrasubbtwn  33055  inagcgrabtwn  33056  iscgrag  33057  cgrgptxfr  33058  leawittrg  33066  rprmasso2  34104  rprmirredlem  34108  rnmptlb  46348  rnmptbddlem  46349  limclner  46754  climisp  46849  climrescn  46851  liminflbuz2  46918  liminflimsupxrre  46920  climxlim2lem  46948  hoicvr  47651