Theorem sbbib 2359

Description: Reversal of substitution. (Contributed by AV, 6-Aug-2023.) (Proof shortened by Wolf Lammen, 4-Sep-2023.)

Hypotheses

Ref	Expression
sbbib.y	⊢ Ⅎ𝑦𝜑
sbbib.x	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
sbbib	⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem sbbib

Step	Hyp	Ref	Expression
1		nfs1v 2153	. . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
2		sbbib.x	. . 3 ⊢ Ⅎ𝑥𝜓
3	1, 2	nfbi 1906	. 2 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 ↔ 𝜓)
4		sbbib.y	. . 3 ⊢ Ⅎ𝑦𝜑
5		nfs1v 2153	. . 3 ⊢ Ⅎ𝑦[𝑥 / 𝑦]𝜓
6	4, 5	nfbi 1906	. 2 ⊢ Ⅎ𝑦(𝜑 ↔ [𝑥 / 𝑦]𝜓)
7		sbequ12r 2245	. . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
8		sbequ12 2244	. . 3 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ [𝑥 / 𝑦]𝜓))
9	7, 8	bibi12d 346	. 2 ⊢ (𝑦 = 𝑥 → (([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ (𝜑 ↔ [𝑥 / 𝑦]𝜓)))
10	3, 6, 9	cbvalv1 2338	1 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))

Syntax hints:   ↔ wb 205  ∀wal 1537  Ⅎwnf 1786  [wsb 2067

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 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068

This theorem is referenced by:  sbbibvv  2360  dfich2  44910