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Theorem cbv3hvOLD 2172
 Description: Obsolete proof of cbv3hv 2171 as of 30-Nov-2020. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 29-Dec-2017.) (Proof shortened by Wolf Lammen, 29-Nov-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
cbv3hv.nf1 (𝜑 → ∀𝑦𝜑)
cbv3hv.nf2 (𝜓 → ∀𝑥𝜓)
cbv3hv.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cbv3hvOLD (∀𝑥𝜑 → ∀𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem cbv3hvOLD
StepHypRef Expression
1 cbv3hv.nf1 . . 3 (𝜑 → ∀𝑦𝜑)
21hbal 2033 . 2 (∀𝑥𝜑 → ∀𝑦𝑥𝜑)
3 ax6ev 1887 . . . . 5 𝑥 𝑥 = 𝑦
4 cbv3hv.1 . . . . 5 (𝑥 = 𝑦 → (𝜑𝜓))
53, 4eximii 1761 . . . 4 𝑥(𝜑𝜓)
6519.35i 1803 . . 3 (∀𝑥𝜑 → ∃𝑥𝜓)
7 cbv3hv.nf2 . . . 4 (𝜓 → ∀𝑥𝜓)
8719.9h 2117 . . 3 (∃𝑥𝜓𝜓)
96, 8sylib 208 . 2 (∀𝑥𝜑𝜓)
102, 9alrimih 1748 1 (∀𝑥𝜑 → ∀𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1478  ∃wex 1701 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044 This theorem depends on definitions:  df-bi 197  df-ex 1702  df-nf 1707 This theorem is referenced by: (None)
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