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Theorem hba1w 1971
 Description: Weak version of hba1 2148. See comments for ax10w 2003. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)
Hypothesis
Ref Expression
hbn1w.1 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
hba1w (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
Distinct variable groups:   𝜑,𝑦   𝜓,𝑥   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem hba1w
StepHypRef Expression
1 hbn1w.1 . . . . . . 7 (𝑥 = 𝑦 → (𝜑𝜓))
21cbvalvw 1966 . . . . . 6 (∀𝑥𝜑 ↔ ∀𝑦𝜓)
32notbii 310 . . . . 5 (¬ ∀𝑥𝜑 ↔ ¬ ∀𝑦𝜓)
43a1i 11 . . . 4 (𝑥 = 𝑦 → (¬ ∀𝑥𝜑 ↔ ¬ ∀𝑦𝜓))
54spw 1964 . . 3 (∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝜑)
65con2i 134 . 2 (∀𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥𝜑)
74hbn1w 1970 . 2 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑)
81hbn1w 1970 . . . 4 (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
98con1i 144 . . 3 (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑)
109alimi 1736 . 2 (∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝑥𝜑)
116, 7, 103syl 18 1 (∀𝑥𝜑 → ∀𝑥𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1478 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 This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1702 This theorem is referenced by: (None)
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