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Theorem rspcebdv 3452
Description: Restricted existential specialization, using implicit substitution in both directions. (Contributed by AV, 8-Jan-2022.)
Hypotheses
Ref Expression
rspcdv.1 (𝜑𝐴𝐵)
rspcdv.2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
rspcebdv.1 ((𝜑𝜓) → 𝑥 = 𝐴)
Assertion
Ref Expression
rspcebdv (𝜑 → (∃𝑥𝐵 𝜓𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝜒,𝑥
Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcebdv
StepHypRef Expression
1 rspcebdv.1 . . . . . . 7 ((𝜑𝜓) → 𝑥 = 𝐴)
2 rspcdv.2 . . . . . . 7 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
31, 2syldan 488 . . . . . 6 ((𝜑𝜓) → (𝜓𝜒))
43biimpd 219 . . . . 5 ((𝜑𝜓) → (𝜓𝜒))
54expcom 450 . . . 4 (𝜓 → (𝜑 → (𝜓𝜒)))
65pm2.43b 55 . . 3 (𝜑 → (𝜓𝜒))
76rexlimdvw 3170 . 2 (𝜑 → (∃𝑥𝐵 𝜓𝜒))
8 rspcdv.1 . . 3 (𝜑𝐴𝐵)
98, 2rspcedv 3451 . 2 (𝜑 → (𝜒 → ∃𝑥𝐵 𝜓))
107, 9impbid 202 1 (𝜑 → (∃𝑥𝐵 𝜓𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1630  wcel 2137  wrex 3049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-v 3340
This theorem is referenced by:  fusgr2wsp2nb  27486
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