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Theorem ralxfrALT 4813
 Description: Alternate proof of ralxfr 4812 which does not use ralxfrd 4805. (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 3278 . . . . 5 (𝐴𝐵 → (∀𝑥𝐵 𝜑𝜓))
41, 3syl 17 . . . 4 (𝑦𝐶 → (∀𝑥𝐵 𝜑𝜓))
54com12 32 . . 3 (∀𝑥𝐵 𝜑 → (𝑦𝐶𝜓))
65ralrimiv 2948 . 2 (∀𝑥𝐵 𝜑 → ∀𝑦𝐶 𝜓)
7 ralxfr.2 . . . 4 (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)
8 nfra1 2925 . . . . 5 𝑦𝑦𝐶 𝜓
9 nfv 1830 . . . . 5 𝑦𝜑
10 rsp 2913 . . . . . 6 (∀𝑦𝐶 𝜓 → (𝑦𝐶𝜓))
112biimprcd 239 . . . . . 6 (𝜓 → (𝑥 = 𝐴𝜑))
1210, 11syl6 34 . . . . 5 (∀𝑦𝐶 𝜓 → (𝑦𝐶 → (𝑥 = 𝐴𝜑)))
138, 9, 12rexlimd 3008 . . . 4 (∀𝑦𝐶 𝜓 → (∃𝑦𝐶 𝑥 = 𝐴𝜑))
147, 13syl5 33 . . 3 (∀𝑦𝐶 𝜓 → (𝑥𝐵𝜑))
1514ralrimiv 2948 . 2 (∀𝑦𝐶 𝜓 → ∀𝑥𝐵 𝜑)
166, 15impbii 198 1 (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175 This theorem is referenced by: (None)
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