Theorem sb4e 2524

Description: One direction of a simplified definition of substitution that unlike sb4b 2499 does not require a distinctor antecedent. Usage of this theorem is discouraged because it depends on ax-13 2390. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.)

Assertion

Ref	Expression
sb4e	⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Proof of Theorem sb4e

Step	Hyp	Ref	Expression
1		sb1 2503	. 2 ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
2		equs5e 2481	. 2 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))
3	1, 2	syl 17	1 ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦 → ∃𝑦𝜑))

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1535  ∃wex 1780  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177  ax-13 2390

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070

This theorem is referenced by:  hbsb2e  2525