Theorem bnj118 34875

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj118.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj118.2	⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)

Assertion

Ref	Expression
bnj118	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Distinct variable groups:   𝐴,𝑛   𝑅,𝑛   𝑓,𝑛   𝑥,𝑛

Allowed substitution hints:   𝜑(𝑥,𝑓,𝑛)   𝐴(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝜑′(𝑥,𝑓,𝑛)

Proof of Theorem bnj118

Step	Hyp	Ref	Expression
1		bnj118.2	. 2 ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)
2		bnj118.1	. . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
3		bnj105 34730	. . 3 ⊢ 1o ∈ V
4	2, 3	bnj91 34867	. 2 ⊢ ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
5	1, 4	bitri 275	1 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Syntax hints:   ↔ wb 206   = wceq 1538  [wsbc 3792  ∅c0 4340  ‘cfv 6566  1_oc1o 8504   predc-bnj14 34694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pow 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-sbc 3793  df-dif 3967  df-un 3969  df-ss 3981  df-nul 4341  df-pw 4608  df-sn 4633  df-suc 6395  df-1o 8511

This theorem is referenced by:  bnj151  34883  bnj153  34886