Theorem spesbcd 3769

Description: form of spsbc 3694. (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 3768	. 2 ⊢ ([𝐴 / 𝑥]𝜓 → ∃𝑥𝜓)
3	1, 2	syl 17	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∃wex 1742  [wsbc 3681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-v 3417  df-sbc 3682

This theorem is referenced by:  euotd  5259  ex-natded9.26  27976  bnj1465  31770  spesbcdi  34848  brtrclfv2  39441  cotrclrcl  39456