Theorem sbbibvv 2392

Description: Reversal of substitution. (Contributed by AV, 6-Aug-2023.)

Assertion

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

Distinct variable groups:   𝜓,𝑥   𝜑,𝑦   𝑥,𝑦

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

Proof of Theorem sbbibvv

Step	Hyp	Ref	Expression
1		nfv 1933	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1933	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	sbbib 2391	1 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))

Syntax hints:   ↔ wb 208  ∀wal 1557  [wsb 2089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-10 2174  ax-11 2190  ax-12 2211

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-nf 1803  df-sb 2090

This theorem is referenced by: (None)