Theorem bnj118 34847

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 34702	. . 3 ⊢ 1o ∈ V
4	2, 3	bnj91 34839	. 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 1537  [wsbc 3804  ∅c0 4352  ‘cfv 6575  1_oc1o 8517   predc-bnj14 34666

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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-pw 4624  df-sn 4649  df-suc 6403  df-1o 8524

This theorem is referenced by:  bnj151  34855  bnj153  34858