Theorem bj-sbftv 32763

Description: Version of sbft 2379 with a dv condition, which does not require ax-13 2246. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-sbftv	⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-sbftv

Step	Hyp	Ref	Expression
1		spsbe 1884	. . 3 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
2		19.9t 2071	. . 3 ⊢ (Ⅎ𝑥𝜑 → (∃𝑥𝜑 ↔ 𝜑))
3	1, 2	syl5ib 234	. 2 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 → 𝜑))
4		nf5r 2064	. . 3 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → ∀𝑥𝜑))
5		bj-stdpc4v 32754	. . 3 ⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
6	4, 5	syl6 35	. 2 ⊢ (Ⅎ𝑥𝜑 → (𝜑 → [𝑦 / 𝑥]𝜑))
7	3, 6	impbid 202	1 ⊢ (Ⅎ𝑥𝜑 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481  ∃wex 1704  Ⅎwnf 1708  [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-nf 1710  df-sb 1881

This theorem is referenced by:  bj-sbfv  32764