Theorem hbsb2e 2491

Description: Special case of a bound-variable hypothesis builder for substitution. Usage of this theorem is discouraged because it depends on ax-13 2377. (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 2490	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
2		sb2 2484	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑) → [𝑦 / 𝑥]∃𝑦𝜑)
3	2	axc4i 2322	. 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑) → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)
4	1, 3	syl 17	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]∃𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1538  ∃wex 1779  [wsb 2064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-10 2141  ax-12 2177  ax-13 2377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065

This theorem is referenced by: (None)