Theorem bj-sbfvv 32765

Description: Version of sbf 2380 with two dv conditions, which does not require ax-10 2019 nor ax-13 2246. (Contributed by BJ, 1-May-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-sbfvv	⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑥

Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem bj-sbfvv

Step	Hyp	Ref	Expression
1		spsbe 1884	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
2		19.9v 1896	. . 3 ⊢ (∃𝑥𝜑 ↔ 𝜑)
3	1, 2	sylib 208	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → 𝜑)
4		ax-5 1839	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
5		bj-stdpc4v 32754	. . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
6	4, 5	syl 17	. 2 ⊢ (𝜑 → [𝑦 / 𝑥]𝜑)
7	3, 6	impbii 199	1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 196  ∀wal 1481  ∃wex 1704  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881

This theorem is referenced by:  bj-vjust2  33015