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Theorem bj-hbs1 32883
Description: Version of hbsb2 2387 with a dv condition, which does not require ax-13 2282, and removal of ax-13 2282 from hbs1 2464. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-hbs1 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-hbs1
StepHypRef Expression
1 bj-sb4v 32882 . 2 ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
2 bj-sb2v 32878 . . 3 (∀𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑)
32axc4i 2169 . 2 (∀𝑥(𝑥 = 𝑦𝜑) → ∀𝑥[𝑦 / 𝑥]𝜑)
41, 3syl 17 1 ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1521  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by:  bj-nfs1v  32884  bj-hbab1  32896
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