Theorem spesbcd 3129

Description: form of spsbc 3059. (Contributed by Mario Carneiro, 9-Feb-2017.)

Hypothesis

Ref	Expression
spesbcd.1	⊢ (φ → [̣A / x]̣ψ)

Assertion

Ref	Expression
spesbcd	⊢ (φ → ∃xψ)

Proof of Theorem spesbcd

Step	Hyp	Ref	Expression
1		spesbcd.1	. 2 ⊢ (φ → [̣A / x]̣ψ)
2		spesbc 3128	. 2 ⊢ ([̣A / x]̣ψ → ∃xψ)
3	1, 2	syl 15	1 ⊢ (φ → ∃xψ)

This proof depends on syntax axioms:   → wi 4  ∃wex 1541  [̣wsbc 3047

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048

This theorem is used by: (None)