Theorem sb6rfv 2345

Description: Reversed substitution. Version of sb6rf 2459 requiring disjoint variables, but fewer axioms. (Contributed by NM, 1-Aug-1993.) (Revised by Wolf Lammen, 7-Feb-2023.)

Hypothesis

Ref	Expression
sb6rfv.nf	⊢ Ⅎ𝑦𝜑

Assertion

Ref	Expression
sb6rfv	⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb6rfv

Step	Hyp	Ref	Expression
1		sb6rfv.nf	. . 3 ⊢ Ⅎ𝑦𝜑
2		sbequ12r 2236	. . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	equsalv 2250	. 2 ⊢ (∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑) ↔ 𝜑)
4	3	bicomi 223	1 ⊢ (𝜑 ↔ ∀𝑦(𝑦 = 𝑥 → [𝑦 / 𝑥]𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1531  Ⅎwnf 1777  [wsb 2059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-nf 1778  df-sb 2060

This theorem is referenced by:  eu1  2598