Theorem reximssdv 3235

Description: Derivation of a restricted existential quantification over a subset (the second hypothesis implies 𝐴 ⊆ 𝐵), deduction form. (Contributed by AV, 21-Aug-2022.)

Hypotheses

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

Assertion

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

Distinct variable group:   𝜑,𝑥

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

Proof of Theorem reximssdv

Step	Hyp	Ref	Expression
1		reximssdv.1	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓)
2		reximssdv.2	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → 𝑥 ∈ 𝐴)
3		reximssdv.3	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → 𝜒)
4	2, 3	jca 515	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → (𝑥 ∈ 𝐴 ∧ 𝜒))
5	4	ex 416	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝜓) → (𝑥 ∈ 𝐴 ∧ 𝜒)))
6	5	reximdv2 3230	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))
7	1, 6	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  ∃wrex 3107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-rex 3112

This theorem is referenced by:  fin1a2lem6  9816  fpwwe2lem12  10052  pgpssslw  18731  efgrelexlemb  18868  lspsneq  19887  lbsextlem4  19926  neissex  21732  iscnp4  21868  nlly2i  22081  llynlly  22082  qtophmeo  22422  ovolicc2lem5  24125  itgsubst  24652  footexALT  26512  footex  26515  opphllem1  26541  lcfl6  38796  mapdval2N  38926  mapdpglem2  38969  hdmaprnlem10N  39155  pellfundglb  39826