Theorem rspcimedv 3568

Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
rspcimdv.1	⊢ (𝜑 → 𝐴 ∈ 𝐵)
rspcimedv.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))

Assertion

Ref	Expression
rspcimedv	⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥   𝜒,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcimedv

Step	Hyp	Ref	Expression
1		rspcimdv.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		rspcimedv.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜒 → 𝜓))
3	2	con3d 152	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (¬ 𝜓 → ¬ 𝜒))
4	1, 3	rspcimdv 3567	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ¬ 𝜓 → ¬ 𝜒))
5	4	con2d 134	. 2 ⊢ (𝜑 → (𝜒 → ¬ ∀𝑥 ∈ 𝐵 ¬ 𝜓))
6		dfrex2 3064	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜓 ↔ ¬ ∀𝑥 ∈ 𝐵 ¬ 𝜓)
7	5, 6	imbitrrdi 252	1 ⊢ (𝜑 → (𝜒 → ∃𝑥 ∈ 𝐵 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062

This theorem is referenced by:  rspcedv  3570  scshwfzeqfzo  14754  symgfixfo  19373  slesolex  22631  usgr2pthlem  29841  clwlkclwwlkfo  30089  satfdmlem  35575