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Theorem ch2var 13058
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 143 . . 3 ((𝑥 = 𝑦𝑧 = 𝑡) → (𝜑𝜓))
51, 2, 42spim 13057 . 2 (∀𝑧𝑥𝜑𝜓)
6 ch2var.min . . 3 𝜑
76ax-gen 1425 . 2 𝑥𝜑
85, 7mpg 1427 1 𝜓
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1329  wnf 1436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1437
This theorem is referenced by:  ch2varv  13059
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