Theorem sbbib 2380

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 2160	. . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
2		sbbib.x	. . 3 ⊢ Ⅎ𝑥𝜓
3	1, 2	nfbi 1904	. 2 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 ↔ 𝜓)
4		sbbib.y	. . 3 ⊢ Ⅎ𝑦𝜑
5		nfs1v 2160	. . 3 ⊢ Ⅎ𝑦[𝑥 / 𝑦]𝜓
6	4, 5	nfbi 1904	. 2 ⊢ Ⅎ𝑦(𝜑 ↔ [𝑥 / 𝑦]𝜓)
7		sbequ12r 2254	. . 3 ⊢ (𝑦 = 𝑥 → ([𝑦 / 𝑥]𝜑 ↔ 𝜑))
8		sbequ12 2253	. . 3 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ [𝑥 / 𝑦]𝜓))
9	7, 8	bibi12d 348	. 2 ⊢ (𝑦 = 𝑥 → (([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ (𝜑 ↔ [𝑥 / 𝑦]𝜓)))
10	3, 6, 9	cbvalv1 2361	1 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))

Syntax hints:   ↔ wb 208  ∀wal 1535  Ⅎwnf 1784  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by:  sbbibvv  2381  dfich2  43662  dfich2ai  43663