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Theorem hbsbv 1833
Description: This is a version of hbsb 1839 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 1662 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
2 ax-17 1435 . . . 4 (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
3 hbsbv.1 . . . 4 (𝜑 → ∀𝑧𝜑)
42, 3hbim 1453 . . 3 ((𝑥 = 𝑦𝜑) → ∀𝑧(𝑥 = 𝑦𝜑))
52, 3hban 1455 . . . 4 ((𝑥 = 𝑦𝜑) → ∀𝑧(𝑥 = 𝑦𝜑))
65hbex 1543 . . 3 (∃𝑥(𝑥 = 𝑦𝜑) → ∀𝑧𝑥(𝑥 = 𝑦𝜑))
74, 6hban 1455 . 2 (((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)) → ∀𝑧((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
81, 7hbxfrbi 1377 1 ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wal 1257  wex 1397  [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  sbco2vlem  1836  2sb5rf  1881  2sb6rf  1882
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