Theorem hbsbv 2001

Description: This is a version of hbsb 2009 with an extra distinct variable constraint, on 𝑧 and 𝑥. (Contributed by Jim Kingdon, 25-Dec-2017.)

Hypothesis

Ref	Expression
hbsbv.1	⊢ (𝜑 → ∀𝑧𝜑)

Assertion

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

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem hbsbv

Step	Hyp	Ref	Expression
1		df-sb 1816	. 2 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
2		ax-17 1579	. . . 4 ⊢ (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦)
3		hbsbv.1	. . . 4 ⊢ (𝜑 → ∀𝑧𝜑)
4	2, 3	hbim 1598	. . 3 ⊢ ((𝑥 = 𝑦 → 𝜑) → ∀𝑧(𝑥 = 𝑦 → 𝜑))
5	2, 3	hban 1600	. . . 4 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ∀𝑧(𝑥 = 𝑦 ∧ 𝜑))
6	5	hbex 1689	. . 3 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑧∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
7	4, 6	hban 1600	. 2 ⊢ (((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) → ∀𝑧((𝑥 = 𝑦 → 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
8	1, 7	hbxfrbi 1525	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑧[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1400  ∃wex 1545  [wsb 1815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-i5r 1588

This theorem depends on definitions:  df-bi 117  df-sb 1816

This theorem is referenced by:  sbco2vlem  2004  2sb5rf  2049  2sb6rf  2050