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Mirrors > Home > MPE Home > Th. List > reximssdv | Structured version Visualization version GIF version |
Description: Derivation of a restricted existential quantification over a subset (the second hypothesis implies 𝐴 ⊆ 𝐵), deduction form. (Contributed by AV, 21-Aug-2022.) |
Ref | Expression |
---|---|
reximssdv.1 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 𝜓) |
reximssdv.2 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → 𝑥 ∈ 𝐴) |
reximssdv.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → 𝜒) |
Ref | Expression |
---|---|
reximssdv | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒) |
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 511 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → (𝑥 ∈ 𝐴 ∧ 𝜒)) |
5 | 4 | ex 412 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝜓) → (𝑥 ∈ 𝐴 ∧ 𝜒))) |
6 | 5 | reximdv2 3161 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 → ∃𝑥 ∈ 𝐴 𝜒)) |
7 | 1, 6 | mpd 15 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ∃wrex 3067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-rex 3068 |
This theorem is referenced by: ttrcltr 9753 fin1a2lem6 10442 fpwwe2lem11 10678 pgpssslw 19646 efgrelexlemb 19782 lspsneq 21141 lbsextlem4 21180 neissex 23150 iscnp4 23286 nlly2i 23499 llynlly 23500 qtophmeo 23840 ovolicc2lem5 25569 itgsubst 26104 footexALT 28740 footex 28743 opphllem1 28769 irngnzply1 33705 weiunfr 36449 lcfl6 41482 mapdval2N 41612 mapdpglem2 41655 hdmaprnlem10N 41841 primrootsunit1 42078 aks6d1c2 42111 aks6d1c6lem5 42158 aks5lem8 42182 pellfundglb 42872 oawordex2 43315 upciclem4 48814 |
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