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Theorem elabreximd 32434
Description: Class substitution in an image set. (Contributed by Thierry Arnoux, 30-Dec-2016.)
Hypotheses
Ref Expression
elabreximd.1 𝑥𝜑
elabreximd.2 𝑥𝜒
elabreximd.3 (𝐴 = 𝐵 → (𝜒𝜓))
elabreximd.4 (𝜑𝐴𝑉)
elabreximd.5 ((𝜑𝑥𝐶) → 𝜓)
Assertion
Ref Expression
elabreximd ((𝜑𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵}) → 𝜒)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑦,𝐶
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem elabreximd
StepHypRef Expression
1 elabreximd.4 . . . 4 (𝜑𝐴𝑉)
2 eqeq1 2730 . . . . . 6 (𝑦 = 𝐴 → (𝑦 = 𝐵𝐴 = 𝐵))
32rexbidv 3169 . . . . 5 (𝑦 = 𝐴 → (∃𝑥𝐶 𝑦 = 𝐵 ↔ ∃𝑥𝐶 𝐴 = 𝐵))
43elabg 3664 . . . 4 (𝐴𝑉 → (𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵} ↔ ∃𝑥𝐶 𝐴 = 𝐵))
51, 4syl 17 . . 3 (𝜑 → (𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵} ↔ ∃𝑥𝐶 𝐴 = 𝐵))
65biimpa 475 . 2 ((𝜑𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵}) → ∃𝑥𝐶 𝐴 = 𝐵)
7 elabreximd.1 . . . 4 𝑥𝜑
8 elabreximd.2 . . . 4 𝑥𝜒
9 simpr 483 . . . . . 6 (((𝜑𝑥𝐶) ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
10 elabreximd.5 . . . . . . 7 ((𝜑𝑥𝐶) → 𝜓)
1110adantr 479 . . . . . 6 (((𝜑𝑥𝐶) ∧ 𝐴 = 𝐵) → 𝜓)
12 elabreximd.3 . . . . . . 7 (𝐴 = 𝐵 → (𝜒𝜓))
1312biimpar 476 . . . . . 6 ((𝐴 = 𝐵𝜓) → 𝜒)
149, 11, 13syl2anc 582 . . . . 5 (((𝜑𝑥𝐶) ∧ 𝐴 = 𝐵) → 𝜒)
1514exp31 418 . . . 4 (𝜑 → (𝑥𝐶 → (𝐴 = 𝐵𝜒)))
167, 8, 15rexlimd 3254 . . 3 (𝜑 → (∃𝑥𝐶 𝐴 = 𝐵𝜒))
1716imp 405 . 2 ((𝜑 ∧ ∃𝑥𝐶 𝐴 = 𝐵) → 𝜒)
186, 17syldan 589 1 ((𝜑𝐴 ∈ {𝑦 ∣ ∃𝑥𝐶 𝑦 = 𝐵}) → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wnf 1778  wcel 2099  {cab 2703  wrex 3060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061
This theorem is referenced by:  elabreximdv  32435  abrexss  32436  iinabrex  32487  disjabrex  32500  disjabrexf  32501
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