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Theorem ralxpxfr2d 3480
Description: Transfer a universal quantifier between one variable with pair-like semantics and two. (Contributed by Stefan O'Rear, 27-Feb-2015.)
Hypotheses
Ref Expression
ralxpxfr2d.a 𝐴 ∈ V
ralxpxfr2d.b (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶𝑧𝐷 𝑥 = 𝐴))
ralxpxfr2d.c ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
ralxpxfr2d (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶𝑧𝐷 𝜒))
Distinct variable groups:   𝜑,𝑥,𝑧   𝜑,𝑦,𝑥   𝜓,𝑦   𝜓,𝑧   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦,𝑧)   𝐴(𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)   𝐶(𝑦,𝑧)   𝐷(𝑦,𝑧)

Proof of Theorem ralxpxfr2d
StepHypRef Expression
1 df-ral 3060 . . . 4 (∀𝑥𝐵 𝜓 ↔ ∀𝑥(𝑥𝐵𝜓))
2 ralxpxfr2d.b . . . . . 6 (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶𝑧𝐷 𝑥 = 𝐴))
32imbi1d 332 . . . . 5 (𝜑 → ((𝑥𝐵𝜓) ↔ (∃𝑦𝐶𝑧𝐷 𝑥 = 𝐴𝜓)))
43albidv 2015 . . . 4 (𝜑 → (∀𝑥(𝑥𝐵𝜓) ↔ ∀𝑥(∃𝑦𝐶𝑧𝐷 𝑥 = 𝐴𝜓)))
51, 4syl5bb 274 . . 3 (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑥(∃𝑦𝐶𝑧𝐷 𝑥 = 𝐴𝜓)))
6 ralcom4 3377 . . . 4 (∀𝑦𝐶𝑥𝑧𝐷 (𝑥 = 𝐴𝜓) ↔ ∀𝑥𝑦𝐶𝑧𝐷 (𝑥 = 𝐴𝜓))
7 ralcom4 3377 . . . . 5 (∀𝑧𝐷𝑥(𝑥 = 𝐴𝜓) ↔ ∀𝑥𝑧𝐷 (𝑥 = 𝐴𝜓))
87ralbii 3127 . . . 4 (∀𝑦𝐶𝑧𝐷𝑥(𝑥 = 𝐴𝜓) ↔ ∀𝑦𝐶𝑥𝑧𝐷 (𝑥 = 𝐴𝜓))
9 r19.23v 3170 . . . . . . 7 (∀𝑧𝐷 (𝑥 = 𝐴𝜓) ↔ (∃𝑧𝐷 𝑥 = 𝐴𝜓))
109ralbii 3127 . . . . . 6 (∀𝑦𝐶𝑧𝐷 (𝑥 = 𝐴𝜓) ↔ ∀𝑦𝐶 (∃𝑧𝐷 𝑥 = 𝐴𝜓))
11 r19.23v 3170 . . . . . 6 (∀𝑦𝐶 (∃𝑧𝐷 𝑥 = 𝐴𝜓) ↔ (∃𝑦𝐶𝑧𝐷 𝑥 = 𝐴𝜓))
1210, 11bitr2i 267 . . . . 5 ((∃𝑦𝐶𝑧𝐷 𝑥 = 𝐴𝜓) ↔ ∀𝑦𝐶𝑧𝐷 (𝑥 = 𝐴𝜓))
1312albii 1914 . . . 4 (∀𝑥(∃𝑦𝐶𝑧𝐷 𝑥 = 𝐴𝜓) ↔ ∀𝑥𝑦𝐶𝑧𝐷 (𝑥 = 𝐴𝜓))
146, 8, 133bitr4ri 295 . . 3 (∀𝑥(∃𝑦𝐶𝑧𝐷 𝑥 = 𝐴𝜓) ↔ ∀𝑦𝐶𝑧𝐷𝑥(𝑥 = 𝐴𝜓))
155, 14syl6bb 278 . 2 (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶𝑧𝐷𝑥(𝑥 = 𝐴𝜓)))
16 ralxpxfr2d.c . . . . . 6 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
1716pm5.74da 838 . . . . 5 (𝜑 → ((𝑥 = 𝐴𝜓) ↔ (𝑥 = 𝐴𝜒)))
1817albidv 2015 . . . 4 (𝜑 → (∀𝑥(𝑥 = 𝐴𝜓) ↔ ∀𝑥(𝑥 = 𝐴𝜒)))
19 ralxpxfr2d.a . . . . 5 𝐴 ∈ V
20 biidd 253 . . . . 5 (𝑥 = 𝐴 → (𝜒𝜒))
2119, 20ceqsalv 3386 . . . 4 (∀𝑥(𝑥 = 𝐴𝜒) ↔ 𝜒)
2218, 21syl6bb 278 . . 3 (𝜑 → (∀𝑥(𝑥 = 𝐴𝜓) ↔ 𝜒))
23222ralbidv 3136 . 2 (𝜑 → (∀𝑦𝐶𝑧𝐷𝑥(𝑥 = 𝐴𝜓) ↔ ∀𝑦𝐶𝑧𝐷 𝜒))
2415, 23bitrd 270 1 (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶𝑧𝐷 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1650   = wceq 1652  wcel 2155  wral 3055  wrex 3056  Vcvv 3350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-v 3352
This theorem is referenced by:  ralxpmap  8112
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