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Theorem 2spim 13003
 Description: Double substitution, as in spim 1716. (Contributed by BJ, 17-Oct-2019.)
Hypotheses
Ref Expression
2spim.nfx 𝑥𝜒
2spim.nfz 𝑧𝜒
2spim.1 ((𝑥 = 𝑦𝑧 = 𝑡) → (𝜓𝜒))
Assertion
Ref Expression
2spim (∀𝑧𝑥𝜓𝜒)
Distinct variable groups:   𝑥,𝑧   𝑥,𝑡
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑡)   𝜒(𝑥,𝑦,𝑧,𝑡)

Proof of Theorem 2spim
StepHypRef Expression
1 2spim.nfz . 2 𝑧𝜒
2 2spim.nfx . . . 4 𝑥𝜒
32a1i 9 . . 3 (𝑧 = 𝑡 → Ⅎ𝑥𝜒)
4 2spim.1 . . . . 5 ((𝑥 = 𝑦𝑧 = 𝑡) → (𝜓𝜒))
54expcom 115 . . . 4 (𝑧 = 𝑡 → (𝑥 = 𝑦 → (𝜓𝜒)))
65alrimiv 1846 . . 3 (𝑧 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → (𝜓𝜒)))
73, 6spimd 13002 . 2 (𝑧 = 𝑡 → (∀𝑥𝜓𝜒))
81, 7spim 1716 1 (∀𝑧𝑥𝜓𝜒)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1329  Ⅎwnf 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:  ch2var  13004
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