Theorem reximssdv 3181

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 521	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → (𝑥 ∈ 𝐴 ∧ 𝜒))
5	4	ex 418	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝜓) → (𝑥 ∈ 𝐴 ∧ 𝜒)))
6	5	reximdv2 3173	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))
7	1, 6	mpd 16	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145  ∃wrex 3087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-rex 3088

This theorem is used by:  ttrcltr  9717  fin1a2lem6  10483  fpwwe2lem11  10726  pgpssslw  19828  efgrelexlemb  19964  lspsneq  21400  lbsextlem4  21439  neissex  23445  iscnp4  23581  nlly2i  23795  llynlly  23796  qtophmeo  24136  ovolicc2lem5  25842  itgsubst  26369  footexALT  29193  footex  29196  opphllem1  29223  irngnzply1  34323  weiunfr  37255  lcfl6  42557  mapdval2N  42687  mapdordlem2  42694  mapdpglem2  42730  hdmaprnlem10N  42916  primrootsunit1  43147  aks6d1c2  43180  aks6d1c6lem5  43227  aks5lem8  43251  pellfundglb  43891  oawordex2  44327  upciclem4  50276