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Theorem hbsbv 1915
 Description: This is a version of hbsb 1923 with an extra distinct variable constraint, on 𝑧 and 𝑥. (Contributed by Jim Kingdon, 25-Dec-2017.)
Hypothesis
Ref Expression
hbsbv.1 (𝜑 → ∀𝑧𝜑)
Assertion
Ref Expression
hbsbv ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem hbsbv
StepHypRef Expression
1 df-sb 1737 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
2 ax-17 1507 . . . 4 (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
3 hbsbv.1 . . . 4 (𝜑 → ∀𝑧𝜑)
42, 3hbim 1525 . . 3 ((𝑥 = 𝑦𝜑) → ∀𝑧(𝑥 = 𝑦𝜑))
52, 3hban 1527 . . . 4 ((𝑥 = 𝑦𝜑) → ∀𝑧(𝑥 = 𝑦𝜑))
65hbex 1616 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑧𝑥(𝑥 = 𝑦𝜑))
74, 6hban 1527 . 2 (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → ∀𝑧((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
81, 7hbxfrbi 1449 1 ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1330  ∃wex 1469  [wsb 1736 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-17 1507  ax-i5r 1516 This theorem depends on definitions:  df-bi 116  df-sb 1737 This theorem is referenced by:  sbco2vlem  1918  2sb5rf  1965  2sb6rf  1966
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