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Theorem ceqsrexbv 2811
Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.)
Hypothesis
Ref Expression
ceqsrexv.1 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsrexbv (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsrexbv
StepHypRef Expression
1 r19.42v 2586 . 2 (∃𝑥𝐵 (𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 ∧ ∃𝑥𝐵 (𝑥 = 𝐴𝜑)))
2 eleq1 2200 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
32adantr 274 . . . . . 6 ((𝑥 = 𝐴𝜑) → (𝑥𝐵𝐴𝐵))
43pm5.32ri 450 . . . . 5 ((𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)))
54bicomi 131 . . . 4 ((𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
65baib 904 . . 3 (𝑥𝐵 → ((𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝑥 = 𝐴𝜑)))
76rexbiia 2448 . 2 (∃𝑥𝐵 (𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ ∃𝑥𝐵 (𝑥 = 𝐴𝜑))
8 ceqsrexv.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
98ceqsrexv 2810 . . 3 (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
109pm5.32i 449 . 2 ((𝐴𝐵 ∧ ∃𝑥𝐵 (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵𝜓))
111, 7, 103bitr3i 209 1 (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wrex 2415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683
This theorem is referenced by:  frecsuclem  6296
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