Theorem sbcth2 3020

Description: A substitution into a theorem. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Mario Carneiro, 13-Oct-2016.)

Hypothesis

Ref	Expression
sbcth2.1	⊢ (𝑥 ∈ 𝐵 → 𝜑)

Assertion

Ref	Expression
sbcth2	⊢ (𝐴 ∈ 𝐵 → [𝐴 / 𝑥]𝜑)

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem sbcth2

Step	Hyp	Ref	Expression
1		sbcth2.1	. . 3 ⊢ (𝑥 ∈ 𝐵 → 𝜑)
2	1	rgen 2507	. 2 ⊢ ∀𝑥 ∈ 𝐵 𝜑
3		rspsbc 3015	. 2 ⊢ (𝐴 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 𝜑 → [𝐴 / 𝑥]𝜑))
4	2, 3	mpi 15	1 ⊢ (𝐴 ∈ 𝐵 → [𝐴 / 𝑥]𝜑)

Syntax hints:   → wi 4   ∈ wcel 2125  ∀wral 2432  [wsbc 2933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-v 2711  df-sbc 2934

This theorem is referenced by: (None)