Theorem bj-hbsb3t 37211

Description: A theorem close to a closed form of hbsb3 2508. (Contributed by BJ, 2-May-2019.)

Assertion

Ref	Expression
bj-hbsb3t	⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))

Proof of Theorem bj-hbsb3t

Step	Hyp	Ref	Expression
1		spsbim 2095	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]∀𝑦𝜑))
2		hbsb2a 2505	. 2 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
3	1, 2	syl6 35	1 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))

Syntax hints:   → wi 4  ∀wal 1548  [wsb 2080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-10 2165  ax-12 2202  ax-13 2393

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-ex 1790  df-nf 1794  df-sb 2081

This theorem is referenced by:  bj-hbsb3  37212  bj-nfs1t  37213