Theorem spesbcd 3849

Description: form of spsbc 3769. (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 3848	. 2 ⊢ ([𝐴 / 𝑥]𝜓 → ∃𝑥𝜓)
3	1, 2	syl 17	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∃wex 1779  [wsbc 3756

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-v 3452  df-sbc 3757

This theorem is referenced by:  euotd  5476  ex-natded9.26  30355  bnj1465  34842  spesbcdi  38121  iscard4  43529  minregex  43530  brtrclfv2  43723  cotrclrcl  43738  rspesbcd  44934