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Theorem ralxfr2d 4461
Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.)
Hypotheses
Ref Expression
ralxfr2d.1 ((𝜑𝑦𝐶) → 𝐴𝑉)
ralxfr2d.2 (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶 𝑥 = 𝐴))
ralxfr2d.3 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
Assertion
Ref Expression
ralxfr2d (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶   𝜒,𝑥   𝜑,𝑥,𝑦   𝜓,𝑦
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐴(𝑦)   𝐶(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem ralxfr2d
StepHypRef Expression
1 ralxfr2d.1 . . . 4 ((𝜑𝑦𝐶) → 𝐴𝑉)
2 elisset 2751 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
31, 2syl 14 . . 3 ((𝜑𝑦𝐶) → ∃𝑥 𝑥 = 𝐴)
4 ralxfr2d.2 . . . . . . . 8 (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶 𝑥 = 𝐴))
54biimprd 158 . . . . . . 7 (𝜑 → (∃𝑦𝐶 𝑥 = 𝐴𝑥𝐵))
6 r19.23v 2586 . . . . . . 7 (∀𝑦𝐶 (𝑥 = 𝐴𝑥𝐵) ↔ (∃𝑦𝐶 𝑥 = 𝐴𝑥𝐵))
75, 6sylibr 134 . . . . . 6 (𝜑 → ∀𝑦𝐶 (𝑥 = 𝐴𝑥𝐵))
87r19.21bi 2565 . . . . 5 ((𝜑𝑦𝐶) → (𝑥 = 𝐴𝑥𝐵))
9 eleq1 2240 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
108, 9mpbidi 151 . . . 4 ((𝜑𝑦𝐶) → (𝑥 = 𝐴𝐴𝐵))
1110exlimdv 1819 . . 3 ((𝜑𝑦𝐶) → (∃𝑥 𝑥 = 𝐴𝐴𝐵))
123, 11mpd 13 . 2 ((𝜑𝑦𝐶) → 𝐴𝐵)
134biimpa 296 . 2 ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)
14 ralxfr2d.3 . 2 ((𝜑𝑥 = 𝐴) → (𝜓𝜒))
1512, 13, 14ralxfrd 4459 1 (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wex 1492  wcel 2148  wral 2455  wrex 2456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739
This theorem is referenced by:  ralrn  5650  ralima  5751  cnrest2  13396  cnptoprest2  13400
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