Theorem reximssdv 3172

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 3164	. 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 3070

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 1913

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1782  df-rex 3071

This theorem is referenced by:  ttrcltr  9713  fin1a2lem6  10402  fpwwe2lem11  10638  pgpssslw  19484  efgrelexlemb  19620  lspsneq  20741  lbsextlem4  20780  neissex  22638  iscnp4  22774  nlly2i  22987  llynlly  22988  qtophmeo  23328  ovolicc2lem5  25045  itgsubst  25573  footexALT  28007  footex  28010  opphllem1  28036  irngnzply1  32815  lcfl6  40457  mapdval2N  40587  mapdpglem2  40630  hdmaprnlem10N  40816  pellfundglb  41705  oawordex2  42158