Theorem bj-hbsb3 37149

Description: Shorter proof of hbsb3 2495. (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 37148	. 2 ⊢ (∀𝑥(𝜑 → ∀𝑦𝜑) → ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑))
2		bj-hbsb3.1	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3	1, 2	mpg 1804	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1545  [wsb 2073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-12 2189  ax-13 2380

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791  df-sb 2074

This theorem is referenced by: (None)