Theorem hbsb2e 2518

Description: Special case of a bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 2404. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
hbsb2e	⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)

Proof of Theorem hbsb2e

Step	Hyp	Ref	Expression
1		sb4e 2517	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
2		sb2 2511	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑) → [𝑦 / 𝑥]∃𝑦𝜑)
3	2	axc4i 2356	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑) → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)
4	1, 3	syl 18	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812  [wsb 2101

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-12 2216  ax-13 2404

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2102

This theorem is used by: (None)