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Theorem ralxfrALT 4227
Description: Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. This proof does not use ralxfrd 4222. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
ralxfr.1 (𝑦𝐶𝐴𝐵)
ralxfr.2 (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)
ralxfr.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ralxfrALT (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓)
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝑥,𝐴   𝑥,𝑦,𝐵   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)   𝐴(𝑦)   𝐶(𝑦)

Proof of Theorem ralxfrALT
StepHypRef Expression
1 ralxfr.1 . . . . 5 (𝑦𝐶𝐴𝐵)
2 ralxfr.3 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
32rspcv 2669 . . . . 5 (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
41, 3syl 14 . . . 4 (𝑦𝐶 → (∀𝑥𝐵 𝜑𝜓))
54com12 30 . . 3 (∀𝑥𝐵 𝜑 → (𝑦𝐶𝜓))
65ralrimiv 2408 . 2 (∀𝑥𝐵 𝜑 → ∀𝑦𝐶 𝜓)
7 ralxfr.2 . . . 4 (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)
8 nfra1 2372 . . . . 5 𝑦𝑦𝐶 𝜓
9 nfv 1437 . . . . 5 𝑦𝜑
10 rsp 2386 . . . . . 6 (∀𝑦𝐶 𝜓 → (𝑦𝐶𝜓))
112biimprcd 153 . . . . . 6 (𝜓 → (𝑥 = 𝐴𝜑))
1210, 11syl6 33 . . . . 5 (∀𝑦𝐶 𝜓 → (𝑦𝐶 → (𝑥 = 𝐴𝜑)))
138, 9, 12rexlimd 2447 . . . 4 (∀𝑦𝐶 𝜓 → (∃𝑦𝐶 𝑥 = 𝐴𝜑))
147, 13syl5 32 . . 3 (∀𝑦𝐶 𝜓 → (𝑥𝐵𝜑))
1514ralrimiv 2408 . 2 (∀𝑦𝐶 𝜓 → ∀𝑥𝐵 𝜑)
166, 15impbii 121 1 (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102   = wceq 1259  wcel 1409  wral 2323  wrex 2324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576
This theorem is referenced by: (None)
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