Theorem bj-hbsb3 34971

Description: Shorter proof of hbsb3 2491. (Contributed by BJ, 2-May-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-hbsb3.1	⊢ (𝜑 → ∀𝑦𝜑)

Assertion

Ref	Expression
bj-hbsb3	⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Proof of Theorem bj-hbsb3

Step	Hyp	Ref	Expression
1		bj-hbsb3t 34970	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
2		bj-hbsb3.1	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3	1, 2	mpg 1800	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1537  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-12 2171  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787  df-sb 2068

This theorem is referenced by: (None)