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Theorem ceqsrexbv 3654
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 3355 . 2 (∃𝑥𝐵 (𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 ∧ ∃𝑥𝐵 (𝑥 = 𝐴𝜑)))
2 eleq1 2905 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
32adantr 481 . . . . . 6 ((𝑥 = 𝐴𝜑) → (𝑥𝐵𝐴𝐵))
43pm5.32ri 576 . . . . 5 ((𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)))
54bicomi 225 . . . 4 ((𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
65baib 536 . . 3 (𝑥𝐵 → ((𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝑥 = 𝐴𝜑)))
76rexbiia 3251 . 2 (∃𝑥𝐵 (𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ ∃𝑥𝐵 (𝑥 = 𝐴𝜑))
8 ceqsrexv.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
98ceqsrexv 3653 . . 3 (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
109pm5.32i 575 . 2 ((𝐴𝐵 ∧ ∃𝑥𝐵 (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵𝜓))
111, 7, 103bitr3i 302 1 (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wrex 3144
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-cleq 2819  df-clel 2898  df-rex 3149
This theorem is referenced by:  marypha2lem2  8889  txkgen  22176  ceqsrexv2  32838  eq0rabdioph  39238
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