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Theorem pgind 48142
Description: Induction on partizan games. (Contributed by Emmett Weisz, 27-May-2024.)
Hypotheses
Ref Expression
pgind.1 (𝑥 = 𝑦 → (𝜓𝜒))
pgind.2 (𝑦 = 𝐴 → (𝜒𝜃))
pgind.3 (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
Assertion
Ref Expression
pgind (𝜑 → (𝐴 ∈ Pg → 𝜃))
Distinct variable groups:   𝑦,𝐴   𝜒,𝑥   𝜓,𝑦   𝜃,𝑦   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑥)   𝐴(𝑥)

Proof of Theorem pgind
StepHypRef Expression
1 19.8a 2170 . 2 (𝜑 → ∃𝑦𝜑)
2 19.8a 2170 . 2 (∃𝑦𝜑 → ∃𝑥𝑦𝜑)
3 nfe1 2140 . . 3 𝑥𝑥𝑦𝜑
4 nfe1 2140 . . . 4 𝑦𝑦𝜑
54nfex 2313 . . 3 𝑦𝑥𝑦𝜑
6 pgind.1 . . 3 (𝑥 = 𝑦 → (𝜓𝜒))
7 pgind.2 . . 3 (𝑦 = 𝐴 → (𝜒𝜃))
8 nfa1 2141 . . . 4 𝑥𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓)
9 nfra1 3277 . . . . . . 7 𝑦𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒
10 nfv 1910 . . . . . . 7 𝑦𝜓
119, 10nfim 1892 . . . . . 6 𝑦(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓)
1211nfal 2312 . . . . 5 𝑦𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓)
13 pgind.3 . . . . 5 (𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
1412, 13exlimi 2206 . . . 4 (∃𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
158, 14exlimi 2206 . . 3 (∃𝑥𝑦𝜑 → ∀𝑥(∀𝑦 ∈ ((1st𝑥) ∪ (2nd𝑥))𝜒𝜓))
163, 5, 6, 7, 15pgindnf 48141 . 2 (∃𝑥𝑦𝜑 → (𝐴 ∈ Pg → 𝜃))
171, 2, 163syl 18 1 (𝜑 → (𝐴 ∈ Pg → 𝜃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1532   = wceq 1534  wex 1774  wcel 2099  wral 3057  cun 3943  cfv 6542  1st c1st 7985  2nd c2nd 7986  Pgcpg 48134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-13 2367  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-1st 7987  df-2nd 7988  df-setrecs 48109  df-pg 48135
This theorem is referenced by: (None)
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