Theorem reximssdv 3205

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 512	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → (𝑥 ∈ 𝐴 ∧ 𝜒))
5	4	ex 413	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝜓) → (𝑥 ∈ 𝐴 ∧ 𝜒)))
6	5	reximdv2 3199	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 → ∃𝑥 ∈ 𝐴 𝜒))
7	1, 6	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  ∃wrex 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-rex 3070

This theorem is referenced by:  ttrcltr  9474  fin1a2lem6  10161  fpwwe2lem11  10397  pgpssslw  19219  efgrelexlemb  19356  lspsneq  20384  lbsextlem4  20423  neissex  22278  iscnp4  22414  nlly2i  22627  llynlly  22628  qtophmeo  22968  ovolicc2lem5  24685  itgsubst  25213  footexALT  27079  footex  27082  opphllem1  27108  lcfl6  39514  mapdval2N  39644  mapdpglem2  39687  hdmaprnlem10N  39873  pellfundglb  40707