Theorem bj-hbsb2av 37164

Description: Version of hbsb2a 2488 with a disjoint variable condition, which does not require ax-13 2376. (Contributed by BJ, 11-Sep-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-hbsb2av	⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-hbsb2av

Step	Hyp	Ref	Expression
1		sb4av 2252	. 2 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		sb6 2092	. . . 4 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
3	2	biimpri 229	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)
4	3	axc4i 2327	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥[𝑦 / 𝑥]𝜑)
5	1, 4	syl 17	1 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1541  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-10 2148  ax-12 2185

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 850  df-ex 1783  df-nf 1787  df-sb 2070

This theorem is referenced by:  bj-hbsb3v  37165