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Theorem nfsb4tALT 2583
Description: Alternate version of nfsb4t 2520. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
dfsb1.p5 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
Assertion
Ref Expression
nfsb4tALT (∀𝑥𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃))

Proof of Theorem nfsb4tALT
StepHypRef Expression
1 dfsb1.p5 . . . . . . . . 9 (𝜃 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
21sbequ12ALT 2560 . . . . . . . 8 (𝑥 = 𝑦 → (𝜑𝜃))
32sps 2183 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → (𝜑𝜃))
43drnf2 2458 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 ↔ Ⅎ𝑧𝜃))
54biimpd 232 . . . . 5 (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝜑 → Ⅎ𝑧𝜃))
65spsd 2185 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∀𝑥𝑧𝜑 → Ⅎ𝑧𝜃))
76impcom 411 . . 3 ((∀𝑥𝑧𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜃)
87a1d 25 . 2 ((∀𝑥𝑧𝜑 ∧ ∀𝑥 𝑥 = 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃))
9 nfnf1 2156 . . . . 5 𝑧𝑧𝜑
109nfal 2334 . . . 4 𝑧𝑥𝑧𝜑
11 nfnae 2448 . . . 4 𝑧 ¬ ∀𝑥 𝑥 = 𝑦
1210, 11nfan 1900 . . 3 𝑧(∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
13 nfa1 2153 . . . 4 𝑥𝑥𝑧𝜑
14 nfnae 2448 . . . 4 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
1513, 14nfan 1900 . . 3 𝑥(∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦)
16 sp 2181 . . . 4 (∀𝑥𝑧𝜑 → Ⅎ𝑧𝜑)
1716adantr 484 . . 3 ((∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑧𝜑)
181nfsb2ALT 2579 . . . 4 (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜃)
1918adantl 485 . . 3 ((∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜃)
202a1i 11 . . 3 ((∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (𝑥 = 𝑦 → (𝜑𝜃)))
2112, 15, 17, 19, 20dvelimdf 2463 . 2 ((∀𝑥𝑧𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃))
228, 21pm2.61dan 812 1 (∀𝑥𝑧𝜑 → (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧𝜃))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1536  wex 1781  wnf 1785
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-10 2143  ax-11 2159  ax-12 2176  ax-13 2382
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786
This theorem is referenced by:  nfsb4ALT  2584
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