Theorem wl-ax11-lem5 37570

Description: Lemma. (Contributed by Wolf Lammen, 30-Jun-2019.)

Assertion

Ref	Expression
wl-ax11-lem5	⊢ (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))

Proof of Theorem wl-ax11-lem5

Step	Hyp	Ref	Expression
1		sbequ12r 2250	. . 3 ⊢ (𝑢 = 𝑦 → ([𝑢 / 𝑦]𝜑 ↔ 𝜑))
2	1	sps 2183	. 2 ⊢ (∀𝑢 𝑢 = 𝑦 → ([𝑢 / 𝑦]𝜑 ↔ 𝜑))
3	2	dral1 2442	1 ⊢ (∀𝑢 𝑢 = 𝑦 → (∀𝑢[𝑢 / 𝑦]𝜑 ↔ ∀𝑦𝜑))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535  [wsb 2062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-10 2139  ax-12 2175  ax-13 2375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063

This theorem is referenced by:  wl-ax11-lem6  37571