Theorem bnj118 34714

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 34569	. . 3 ⊢ 1o ∈ V
4	2, 3	bnj91 34706	. 2 ⊢ ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
5	1, 4	bitri 274	1 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))

Syntax hints:   ↔ wb 205   = wceq 1534  [wsbc 3776  ∅c0 4325  ‘cfv 6554  1_oc1o 8489   predc-bnj14 34533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-pw 4609  df-sn 4634  df-suc 6382  df-1o 8496

This theorem is referenced by:  bnj151  34722  bnj153  34725