Theorem sbbib 2367

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 2157	. . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
2		sbbib.x	. . 3 ⊢ Ⅎ𝑥𝜓
3	1, 2	nfbi 1902	. 2 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 ↔ 𝜓)
4		sbbib.y	. . 3 ⊢ Ⅎ𝑦𝜑
5		nfs1v 2157	. . 3 ⊢ Ⅎ𝑦[𝑥 / 𝑦]𝜓
6	4, 5	nfbi 1902	. 2 ⊢ Ⅎ𝑦(𝜑 ↔ [𝑥 / 𝑦]𝜓)
7		sbequ12r 2253	. . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
8		sbequ12 2252	. . 3 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ [𝑥 / 𝑦]𝜓))
9	7, 8	bibi12d 345	. 2 ⊢ (𝑦 = 𝑥 → (([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ (𝜑 ↔ [𝑥 / 𝑦]𝜓)))
10	3, 6, 9	cbvalv1 2347	1 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))

Syntax hints:   ↔ wb 206  ∀wal 1535  Ⅎwnf 1781  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-10 2141  ax-11 2158  ax-12 2178

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-nf 1782  df-sb 2065

This theorem is referenced by:  sbbibvv  2368  dfich2  47332