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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 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) |
Colors of variables: wff setvar class |
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 |
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