Theorem hbsb2a 2514

Description: Special case of a bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 2402. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
hbsb2a	⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Proof of Theorem hbsb2a

Step	Hyp	Ref	Expression
1		sb4a 2510	. 2 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥(𝑥 = 𝑦 → 𝜑))
2		sb2 2509	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → [𝑦 / 𝑥]𝜑)
3	2	axc4i 2353	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) → ∀𝑥[𝑦 / 𝑥]𝜑)
4	1, 3	syl 18	1 ⊢ ([𝑦 / 𝑥]∀𝑦𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)

Syntax hints:   → wi 4  ∀wal 1566  [wsb 2094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-10 2174  ax-12 2211  ax-13 2402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1808  df-nf 1812  df-sb 2095

This theorem is referenced by:  hbsb3  2517  bj-hbsb3t  37389