Theorem sbbibvv 2396

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 1937	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1937	. 2 ⊢ Ⅎ𝑥𝜓
3	1, 2	sbbib 2395	1 ⊢ (∀𝑦([𝑦 / 𝑥]𝜑 ↔ 𝜓) ↔ ∀𝑥(𝜑 ↔ [𝑥 / 𝑦]𝜓))

Syntax hints:   ↔ wb 209  ∀wal 1561  [wsb 2093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-10 2178  ax-11 2194  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-nf 1807  df-sb 2094

This theorem is referenced by: (None)