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Theorem ch2var 9907
 Description: Implicit substitution of 𝑦 for 𝑥 and 𝑡 for 𝑧 into a theorem. (Contributed by BJ, 17-Oct-2019.)
Hypotheses
Ref Expression
ch2var.nfx 𝑥𝜓
ch2var.nfz 𝑧𝜓
ch2var.maj ((𝑥 = 𝑦𝑧 = 𝑡) → (𝜑𝜓))
ch2var.min 𝜑
Assertion
Ref Expression
ch2var 𝜓
Distinct variable groups:   𝑥,𝑧   𝑥,𝑡
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑡)   𝜓(𝑥,𝑦,𝑧,𝑡)

Proof of Theorem ch2var
StepHypRef Expression
1 ch2var.nfx . . 3 𝑥𝜓
2 ch2var.nfz . . 3 𝑧𝜓
3 ch2var.maj . . . 4 ((𝑥 = 𝑦𝑧 = 𝑡) → (𝜑𝜓))
43biimpd 132 . . 3 ((𝑥 = 𝑦𝑧 = 𝑡) → (𝜑𝜓))
51, 2, 42spim 9906 . 2 (∀𝑧𝑥𝜑𝜓)
6 ch2var.min . . 3 𝜑
76ax-gen 1338 . 2 𝑥𝜑
85, 7mpg 1340 1 𝜓
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241  Ⅎwnf 1349 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427 This theorem depends on definitions:  df-bi 110  df-nf 1350 This theorem is referenced by:  ch2varv  9908
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