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Theorem ceqsrexbv 2698
 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 2484 . 2 (∃𝑥𝐵 (𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 ∧ ∃𝑥𝐵 (𝑥 = 𝐴𝜑)))
2 eleq1 2116 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
32adantr 265 . . . . . 6 ((𝑥 = 𝐴𝜑) → (𝑥𝐵𝐴𝐵))
43pm5.32ri 436 . . . . 5 ((𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)))
54bicomi 127 . . . 4 ((𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
65baib 839 . . 3 (𝑥𝐵 → ((𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝑥 = 𝐴𝜑)))
76rexbiia 2356 . 2 (∃𝑥𝐵 (𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ ∃𝑥𝐵 (𝑥 = 𝐴𝜑))
8 ceqsrexv.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
98ceqsrexv 2697 . . 3 (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
109pm5.32i 435 . 2 ((𝐴𝐵 ∧ ∃𝑥𝐵 (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵𝜓))
111, 7, 103bitr3i 203 1 (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   = wceq 1259   ∈ wcel 1409  ∃wrex 2324 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576 This theorem is referenced by:  frecsuclem3  6021
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