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Theorem ralxfrALT 5285
 Description: Alternate proof of ralxfr 5284 which does not use ralxfrd 5278. (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 3567 . . . . 5 (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
41, 3syl 17 . . . 4 (𝑦𝐶 → (∀𝑥𝐵 𝜑𝜓))
54com12 32 . . 3 (∀𝑥𝐵 𝜑 → (𝑦𝐶𝜓))
65ralrimiv 3148 . 2 (∀𝑥𝐵 𝜑 → ∀𝑦𝐶 𝜓)
7 ralxfr.2 . . . 4 (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)
8 nfra1 3183 . . . . 5 𝑦𝑦𝐶 𝜓
9 nfv 1915 . . . . 5 𝑦𝜑
10 rsp 3170 . . . . . 6 (∀𝑦𝐶 𝜓 → (𝑦𝐶𝜓))
112biimprcd 253 . . . . . 6 (𝜓 → (𝑥 = 𝐴𝜑))
1210, 11syl6 35 . . . . 5 (∀𝑦𝐶 𝜓 → (𝑦𝐶 → (𝑥 = 𝐴𝜑)))
138, 9, 12rexlimd 3277 . . . 4 (∀𝑦𝐶 𝜓 → (∃𝑦𝐶 𝑥 = 𝐴𝜑))
147, 13syl5 34 . . 3 (∀𝑦𝐶 𝜓 → (𝑥𝐵𝜑))
1514ralrimiv 3148 . 2 (∀𝑦𝐶 𝜓 → ∀𝑥𝐵 𝜑)
166, 15impbii 212 1 (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112 This theorem is referenced by: (None)
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