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Theorem ceqsrexbv 3476
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 3230 . 2 (∃𝑥𝐵 (𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 ∧ ∃𝑥𝐵 (𝑥 = 𝐴𝜑)))
2 eleq1 2827 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
32adantr 472 . . . . . 6 ((𝑥 = 𝐴𝜑) → (𝑥𝐵𝐴𝐵))
43pm5.32ri 673 . . . . 5 ((𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)))
54bicomi 214 . . . 4 ((𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝑥𝐵 ∧ (𝑥 = 𝐴𝜑)))
65baib 982 . . 3 (𝑥𝐵 → ((𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ (𝑥 = 𝐴𝜑)))
76rexbiia 3178 . 2 (∃𝑥𝐵 (𝐴𝐵 ∧ (𝑥 = 𝐴𝜑)) ↔ ∃𝑥𝐵 (𝑥 = 𝐴𝜑))
8 ceqsrexv.1 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
98ceqsrexv 3475 . . 3 (𝐴𝐵 → (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ 𝜓))
109pm5.32i 672 . 2 ((𝐴𝐵 ∧ ∃𝑥𝐵 (𝑥 = 𝐴𝜑)) ↔ (𝐴𝐵𝜓))
111, 7, 103bitr3i 290 1 (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wrex 3051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-12 2196  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-rex 3056  df-v 3342
This theorem is referenced by:  marypha2lem2  8509  txkgen  21677  ceqsrexv2  31933  eq0rabdioph  37860
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