Theorem sb9i 2523

Description: Commutation of quantification and substitution variables. Usage of this theorem is discouraged because it depends on ax-13 2375. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 15-Jun-2019.) (New usage is discouraged.)

Assertion

Ref	Expression
sb9i	⊢ (∀𝑥[𝑥 / 𝑦]𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)

Proof of Theorem sb9i

Step	Hyp	Ref	Expression
1		sb9 2522	. 2 ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 ↔ ∀𝑦[𝑦 / 𝑥]𝜑)
2	1	biimpi 216	1 ⊢ (∀𝑥[𝑥 / 𝑦]𝜑 → ∀𝑦[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1537  [wsb 2063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064

This theorem is referenced by: (None)