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Theorem rexrsb 41675
Description: An equivalent expression for restricted existence, analogous to exsb 2605. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
Assertion
Ref Expression
rexrsb (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝑥 = 𝑦𝜑))
Distinct variable groups:   𝑥,𝑦,𝐴   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rexrsb
StepHypRef Expression
1 rexsb 41674 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥(𝑥 = 𝑦𝜑))
2 alral 3066 . . . 4 (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥𝐴 (𝑥 = 𝑦𝜑))
3 df-ral 3055 . . . . . 6 (∀𝑥𝐴 (𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥𝐴 → (𝑥 = 𝑦𝜑)))
4 19.27v 2073 . . . . . . . 8 (∀𝑥((𝑥𝐴 → (𝑥 = 𝑦𝜑)) ∧ 𝑦𝐴) ↔ (∀𝑥(𝑥𝐴 → (𝑥 = 𝑦𝜑)) ∧ 𝑦𝐴))
5 pm2.04 90 . . . . . . . . . . 11 ((𝑥𝐴 → (𝑥 = 𝑦𝜑)) → (𝑥 = 𝑦 → (𝑥𝐴𝜑)))
6 eleq1w 2822 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
76biimprd 238 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑦𝐴𝑥𝐴))
8 pm2.83 84 . . . . . . . . . . . 12 ((𝑥 = 𝑦 → (𝑦𝐴𝑥𝐴)) → ((𝑥 = 𝑦 → (𝑥𝐴𝜑)) → (𝑥 = 𝑦 → (𝑦𝐴𝜑))))
97, 8ax-mp 5 . . . . . . . . . . 11 ((𝑥 = 𝑦 → (𝑥𝐴𝜑)) → (𝑥 = 𝑦 → (𝑦𝐴𝜑)))
10 pm2.04 90 . . . . . . . . . . 11 ((𝑥 = 𝑦 → (𝑦𝐴𝜑)) → (𝑦𝐴 → (𝑥 = 𝑦𝜑)))
115, 9, 103syl 18 . . . . . . . . . 10 ((𝑥𝐴 → (𝑥 = 𝑦𝜑)) → (𝑦𝐴 → (𝑥 = 𝑦𝜑)))
1211imp 444 . . . . . . . . 9 (((𝑥𝐴 → (𝑥 = 𝑦𝜑)) ∧ 𝑦𝐴) → (𝑥 = 𝑦𝜑))
1312alimi 1888 . . . . . . . 8 (∀𝑥((𝑥𝐴 → (𝑥 = 𝑦𝜑)) ∧ 𝑦𝐴) → ∀𝑥(𝑥 = 𝑦𝜑))
144, 13sylbir 225 . . . . . . 7 ((∀𝑥(𝑥𝐴 → (𝑥 = 𝑦𝜑)) ∧ 𝑦𝐴) → ∀𝑥(𝑥 = 𝑦𝜑))
1514ex 449 . . . . . 6 (∀𝑥(𝑥𝐴 → (𝑥 = 𝑦𝜑)) → (𝑦𝐴 → ∀𝑥(𝑥 = 𝑦𝜑)))
163, 15sylbi 207 . . . . 5 (∀𝑥𝐴 (𝑥 = 𝑦𝜑) → (𝑦𝐴 → ∀𝑥(𝑥 = 𝑦𝜑)))
1716com12 32 . . . 4 (𝑦𝐴 → (∀𝑥𝐴 (𝑥 = 𝑦𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
182, 17impbid2 216 . . 3 (𝑦𝐴 → (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥𝐴 (𝑥 = 𝑦𝜑)))
1918rexbiia 3178 . 2 (∃𝑦𝐴𝑥(𝑥 = 𝑦𝜑) ↔ ∃𝑦𝐴𝑥𝐴 (𝑥 = 𝑦𝜑))
201, 19bitri 264 1 (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐴𝑥𝐴 (𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630  wcel 2139  wral 3050  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-10 2168  ax-11 2183  ax-12 2196  ax-13 2391
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-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056
This theorem is referenced by:  2rexrsb  41677
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